[Composite reply]
My grandmother, mother and so consequently so do I (it was how I was taught) always used to flour mince when it came to adding it to the pan.
Now that is interesting; that suggests to me that it may be analogous to "velveting" chicken, something I learned from my (Oriental) wife.
I assume the onions are being fried in oil or butter. The flour will act as a thickener when added to the fat. The process, as you must know, is called roux. Then, depending on how dark you want the sauce, you brown the oil/butter and flour to the level required.
OK, but that's close to "name magic" — saying that it creates a roux doesn't explain why one might
want/need to create a roux — is it for thickening, is it for colour, is it for flavour, etc. ?
I have a cookbook from renowned Cajun / Louisiana chef Paul Prudhomme. (Bought for me 20 years ago by the woman who is now the current Mrs Robbo, before we were a thing, but that’s another story). In this book, he has several pictures of roux in varying degrees of colour, from a fairly light mustardy to a dark brown, depending on the dish being cooked. He explains in great detail why the roux is cooked to these different stages for various dishes such as jambalaya, gumbo etc.
Interesting — as I lack M. Prudhomme's book, I shall consult
Practical Professional Cookery (Cracknell and Kaufmann) and see what it has to say about rouxs. I shall also ask our (French) head chef if he has the book to which you refer.
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** Phil.