How do I fix a sauce that has broken?

Well, part of my quandary with regards to thickening the sauce was whether to thicken to taste or viscosity. It would be helpful for me to know the science behind various thickening mediums/liquids. I understand how starches can thicken a sauce but I don't really know if milk will thicken like cream (once the curds are strained and whisked) or if I needed to add something to it to thicken it if I wanted it to remain sauce-like and not just reduce it all the way down to little nuggets of curd. I reduced it well beyond what the recipe called for and it was tasty but still thinner than I wanted.

Also, I generally reach for flour as a thickener b/c that's how I learned (from my mom), but the jury's still out for me. I use cornstarch in Asian cuisines as well and agree with you that the consistency can be kind of gummy.

NeuB, take your insults somewhere else. We don't want or need them here. Sam1148 gives great answers, has tons of knowledge on the subjects he answers, and NEVER ONCE have I heard him judge anyone for their lack of experience or for asking 'silly' questions.

I'm not trying to add fuel to the fire for the argument on how to thicken a sauce, but you mentioned that your sauce was a bit thin. When roasting meat and making a pan sauce, it's often best to reduce the sauce further as a means of thickening it rather than adding starch as a thickener (though the flour+butter idea is a pretty traditional way to thicken a sauce and might have worked for your situation). You can also whisk in a little more butter or cream (which is an emulsifier) at the end to help thicken, though your sauce will break if the temperature of the pan is too high (it will fall out of emulsion once it begins to boil). I happen to dislike using cornstarch to thicken most sauces outside of those used for stir fries or in Asian cuisines; it can impart an artificially gelatinous and gummy texture that doesn't always suit a pan sauce.

I believe that Christina Tosi in Momofuku Milk Bar says you can use heavy cream to fix a broken sauce.

Actually, this version calls for the butter at the beginning of braising, not the end, soI didn't have time to start the sauce over once it broke. (it was after a couple of hours -- I turned up the heat out of impatience. But I'll try a slurry next time. Thanks.

Sam1148, yes, it's my first time posting and I posted a more detailed question a while before, s
then thought it was too wordy and reposted succinctly but neglected to explain further below! Now I've got the format! OK, so it was pork braised in milk. This version had a little butter, and the liquid reduced only by about 2/3 (and the curds pushed through a strainer), so that the sauce was quite thin instead of winding up as little nuggets of curd as in other versions. Cooking vessel was an enameled cast iron Dutch oven.

In another bowl, add a small amount (1 tbs) of one of the ingredients that caused sauce to break (oil, egg, etc). Very slowly, whisk brOken sauce into new bowl

You need to expand with some more details here. What type of sauce..etc.

As a general answer, for generic 'sauce'. Reheat the sauce.
In the microwave, add some flour and butter..and heat. add a bit of the broken sauce to that bowl. Wisk and heat that surry back into the broken sauce. repeat as needed.

Too many variables here to give a hard answer without knowing the type of the sauce, cookware, and ultimate goal.