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 Troy pound) cotton and 3 ochaus (eighths ie. 1/8 Troy 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 Troy pound and 1 Troy pound 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.
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.
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. 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 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.
 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