What is the thickening power of different types of roux?

My understanding is that the ratio shifts from 10:1 (liquid to roux) for a pale roux, up to about 5:1 for a nutty brown roux, with a somewhat linear relationship between the two extremes. A dark brown roux has very little thickening power, mostly they are used for flavor more than actual work-a-day thickening.

The problem may not so much in the roux ratios, but in other ingredient variations. For instance, if you are using a traditional 3:2 flour:fat roux, and your fat is butter, make sure you are using clarified butter, or that you are allowing most of the water to cook out of the butter before adding the flour - excess water in the roux will break down some of the starches and reduce the thickening strength.

Finally, there will be some variation no matter what - you can always make a little extra roux in a second pan to integrate if necessary.

I don't know if it's possible to give a very accurate roux:liquid ratio as a function of color because it can be hard to accurately and repeatably judge the color. Your best bet is to add some of the liquid -- maybe half or so of what you expect to need eventually, whisk until smooth, and bring it to a boil. It should thicken up at that point, and then you can start adding more of the liquid until you get to the consistency you want. Remember that the sauce will thicken a bit more as you cook it due to evaporation, and it also seems to thicken a bit as it cools. So if you start with it just a touch on the thin side, it'll probably be perfect at serving time.

Maybe there does exist a roux color chart that can help, or maybe you're not looking for super accurate numbers. After checking "Cooking" (Peterson), "The New Professional Chef" (Culinary Institute of America), "On Food and Cooking" (McGee), and "Cookwise" (Corriher), the only guideline on ratio I found was from Shirly Corriher, who tells us that you want 1 tablespoon of flour per cup of liquid for a thin sauce, 2 tablespoons per cup for a medium sauce, and 3 tablespoons per cup for a thick sauce. There's a discussion of why darker roux thickens less, of course, but no attempt to quantify that effect. Corriher goes on to point out other variables: age of the flour, protein content (higher protein flour thickens less), and other ingredients (salt, sugar, acid).

Perhaps because there are so many variables that are hard to quantify and control, this is one of those areas where most cooks rely on a little experience and a lot of observation.

Try here- https://www.inkling.com/read/professional-chef-cia-9th/chapter-11/roux

Although this doesn't specifically answer the question, it does include a comparative colour chart for roux, which will be most helpful in gaining a consistent thickening power.

From there I'd be tempted to make a batch of sauce as suggested above, using a given quantity of roux at the correct colour & adding stock to derive a known quantity of stock to roux. Thereby giving the ratio you'll use later.

Once noted down it should be easy to derive a ratio and make-up the rest of the sauce with a new roux, and you'll know your numbers for next time. :)