Three of the four surviving globulose-breasted garments are heavy overgarments for soldiers, while the only one meant to be worn next to the shirt (the gold Charles de Blois garment) just has a layer of cloth of gold, a few millimetres of cotton wool, and a layer of fine linen. How typical was that? Was it a light summer garment, or did Charles de Blois just happen to have the body shape which was in fashion?
We are fortunate that in the late middle ages and renaissance, tailors were just paid for labour. The customer told them what they wanted and what kind of cloth they intended to use, the tailor quoted them a yardage of each fabric, and the customer provided it and collected the finished garment and the scraps when the project was done. About 90% of the cost of clothing was the fabric and notions. This means that medieval account books routinely list the materials used to make specific garments. I have not found very many from the period I am interested in, but the ones from the era of the ‘snaky profile’ or ‘weasel waist’ in the 1340s are suggestive.
The brothers Bonis were merchants of Montalban in Languedoc. In September 1345 Gualhart de Guordo, a burger, bought 4 ells and a palm of fustian at 3 s. the ell, 3 ells and a palm of linen at 12 d. the ell, 3 quarters (ie. 3/4 of a Troyes pound of 14 medieval ounces) cotton and 3 ochaus (eighths ie. 3/8 of a Troyes pound) of silk to send to his tailor to make a jupon, total 20 s. 1 d. (part 1, p. 180, fol. 37). In other sales in the 1340s, their customers used anything between 1/2 Troyes pound (about 217 g) and 1 Troyes pound (about 434 g) of cotton wool in a jupon, normally quilted between a layer of fustian and a layer of linen. You can find part 1 on the Internet Archive https://archive.org/stream/leslivresdecompt01boni#page/n95/mode/2up and part 2 on Gallica http://gallica.bnf.fr/ark:/12148/bpt6k102257q
Edward III’s Great Wardrobe accounts from the 1340s were published in 1846. His armourer used between 1 and 2 pounds of cotton to make a doublet (including one arming doublet), and up to 3 pounds in the heaviest garments for men. A typical entry goes like this: “And for making one aketon for the King covered with blue satin: 4 ells of satin, 4 ells and a half of linen of Paris, 4 ells and a half of linen of Rheims, 2 pounds of cotton, a half quarter (ie. 1/8 pound) of silk, 1 ell reinforced sindon.” (Nicholas, “Observations on the Order of the Garter,” p. 49). You can find the Latin on Armour in Texts.
It is hard to prove which ell and which pound the clerks were using, but we can compare similar garments made out of different materials, and study evidence for the widths of different fabrics. European fabrics containing cotton, for example, were usually about 50-60 cm wide, while wool broadcloth in this period was probably three times as wide. Based on this evidence, I suspect that Edward III’s clerks used an ell no more than 90 cm/36″ long, and that the satin, linen of Paris, and linen of Rheims had about the same area, so that this aketon consisted of a layer of satin, a layer of linen, a layer of raw cotton, and a layer of linen. This would be similar to the surviving pourpoint of Charles de Blois and farsetto of Pandolfo III Malatesta, and give enough 20-24″ wide satin to make a knee-length garment as was fashionable in the 1340s.
The rules of the armourers of Paris from 1311 mentioned two types of quilted coat: one which must contain at least three pounds of cotton, and one which must contain at least six. I suspect that these are overcoats, like the jack of late-15th-century and early-16th-century sources. You can find the French on Armour in Texts. The accounts of Robert of Artois from 1314 contain an entry for an aketon with three pounds of stuffing: “for a pound of silk bourre for Robert’s aketon, 6 s.; for 2 pounds of cotton to put in the said aketon with the aforesaid bourre, 4 s.; 5 ells of white linen for Robert’s aketon, price 2 s. 6 d. the ell, 12 s. 6 d.” 
Doublets could be even lighter. In 1393/4 someone in Roger Mortimer’s household bought 2 ells of linen cloth at 8 1/2d. the ell “for a doublet for dancing,” and 2 ells of linen cloth for lining the same doublet at 11d. the ell (Baildon “A Wardrobe Account of 16-17 Richard II, 1393-4” p. 513). After looking at art from the day, I suspect that the doublet still gave its wearer the fashionable profile.
At least one modern medieval tailor makes shaped pillows to put inside the breast of a doublet. While this is a clever idea, I do not know of any text or any illustration which supports it.
There were certainly some very heavy garments stuffed with cotton. Documents from Venice from the end of the 13th century say that infantry should wear overgarments stuffed with 8 libbre (probably the libbra grossa of 477 g) of cotton and bearing the sign of St. Mark (Cessi, Deliberazioni del Maggior Consiglio di Venezia, III, p. 17, 406). Tasha Kelly estimates that the red brocade garment of Charles VI would have weighed 10 to 12 pounds if it had been made for a grown man. But I think that those related to the quilted garments worn next to the shirt like the Burgundian jacks of 20 or 30 linens compared to the Burgundian “pourpoint for wearing under armour” which was made “of damask cloth (a patterned silk), lined (doublé) with six linens and a blanket (ie. a cheap (-et) white (blanc) wool cloth suitable for covering beds)” in 1432/3. Heavy garments created the same shapes as light garments, just using more layers or more stuffing. The fashionable silhouette for any given decade still had to be one which a tailor could create with a light garment.
On the other hand, medieval tailors usually required a quarter or an eighth of a pound of silk (= heavy thread like modern buttonhole silk?) and often several ounces of linen thread to make a pourpoint, jupon, aketon, or doublet, and it was a commonplace that the more stitches the better the pourpoint. So I think we should study the techniques used by tailors to create lightweight garments which stand away from the body in some spots and follow it tightly in others. Jessica Finley of Fühlen Designs and Matthew Gnagy of the Modern Maker are experimenting with those techniques, and I am sure they are not alone.
Edit 2018-02-25: changed the description of the damask doublet from a paraphrase to a translation.
Edit 2018-10-02: added a short paragraph on the construction which I think the Great Wardrobe Accounts have in mind
Edit 2018-10-06: rethought my statement of the length of the ell in the Great Wardrobe accounts … it is possible that this ell is shorter than the yard of later sources (Latin documents don’t always distinguish between yards and ells)
Edit 2019-01-22: Added the entry in the accounts of Robert of Artois.
Edit 2019-02-18: Clarified that it was pound of Troyes not the English Troy pound which Montalban adopted in 1329 (Forestie vol. 1 pp. xxxviii, xxxix). The English word may be related to German <treu> “loyal, faithful, trustworthy” not the city in France: for the full story, see Alfred E. Lieber, “Barley Coins and Troy Grains. The Evolution of the English system of Weights,” in Jean-Claude Hocquet (ed.), Une Activitie Universelle. Peser et mesurer à travers les âges. Cahiers de Métrologie tome 11-12. Acta Metrologiae IV: VIe Congres International de Metrologie Historique (Editions-Diffusion du Lys: Caen, 1994) pp. 433-440
Edit 2019-03-27: Added the Venetian documents from the end of the 13th century; I have not checked the citation against the original publication yet!
 For examples of the ratio between the price of clothing and the price of tailoring, see Textiles and Clothing, The King’s Servants, or Drei Schnittbücher. On underwear the price of tailoring could rise as high as 25% or so, since low-end linens and canvas were the cheapest fabrics.
 L.E.S.J. Laborde, Ducs de Bourgogne: études sur les lettres, les artes et l’industrie pendant le XVe siècle et plus particulèrement dans le Pays-Bas et le duché de Bourgogne, Tome 1 p. 310 (1432/3): “Item pour la façon d’un autre pourpoint de drap de damas doublé de VI toilles et de ung blanchet pour armer dessus, XL s. – Item qu’il a paié pour VIII aulnes de toille, au pris de X blancs l’aulne, font XXVI s. VIII d. – Item qu’il a aussi paié pour demie aulne de blanchet pour mettre dedans l’un de ses pourpoins, VIII s.” http://gallica.bnf.fr/ark:/12148/bpt6k164542d/f457.image.rch=laborde+l%C3%A9on.langFR
 Tasha Kelly-Mele estimates that the pourpoint of Charles de Blois was covered with a piece of cloth 2.70 x 50-55 cm (3 American yards by 20-21″). This garment was made a generation later than the 1340s when hemlines had risen to just below the crotch, so less fabric was required. There is some dispute about the width of silk cloth in the middle of the 14th century, but merchants’ guides, tailor’s handbooks, and surviving pieces from the 16th century show that by that period silk cloth was usually around 50-60 cm wide. Assuming that silk cloth in 14th century England had similar widths, about 1/3 the width of wool broadcloth, gives plausible results and is consistent with one garment from the period.
 “1314.- Pour une livre de boure de soie pour l’auqueton Robert 6 s..- Pour 2 l. de coton mis audit auqueton avoec ladite boure, 4 s.; 5 aunes de blanc toile pour l’auqueton Robert, 2 s. 6 d. l’aune, 12 s. 6 d. (Cpte de l’hotel de Robert d’Artois; Arch. du Pas-de-Calais)” (Gay s.v. hoqueton)