Here is how people did it in the rural backwoods of Southwest Washington State:
You take the cast iron pan in which you have baked your meat for the meal, and it will have at the bottom grease drippings plus lots of brown bits. You put that pan on a burner something like medium heat, and stir in your flour stirring constantly until the flour is cooked. You get so you can just tell how much flour to put in so the fat is absorbed and the gravy is not greasy. It is the same thing as a roux, although we never called it that.
Then there is something I do not quite remember...maybe a better cook will come to your question...but you have to do something so when you add the milk it does not make the gravy lumpy. I think maybe it was to add the milk a little at a time, stirring constantly. And you get so you can also tell just how much milk to add so the gravy is just the right thickness.
Then adjust seasoning if needed. It is simple and very delicious, and you do it at the very last minute so the gravy is hot and fresh.
To make brown gravy from flour you will need to cook the flour until it turns brown. Put about a teaspoon of grease in your pan and add your flour and cook until it's brown then add your milk or cream and cook until it is thickened.
Usually, you would have drippings from cooking a roast, chicken, ot turkey. Then you would stir in a couple of tablespoons of flour to thicken it while stirring it constantly over low heat until it is the consistency you like.
I've never heard milk in gravy, only in mashed potatoes. I use the very pan I cooked the meat in and I scrape the pan to get everything I can into the gravy. The secret is to add the flour gradually, but before you add it, in a small bowl, put the dry flour and add some of your hot drippings and mush the flour into a paste, then add more drippings to thin it down, then blend that into your pan. If you add the dry flour to the pan, it will most likely lump up. If you like thick gravy, add more flour. The gravy must come to a boil to thicken. I use about 4 tablespoons of flour for two cups of gravy. Very thick!
My grandma used to make brown flour gravy for when we were sick with stomach bugs. She used to brown just flour in a cast iron pan, and then add milk. She said that it used to calm your tummy down when you had any kind of stomach flu or what have you. That's what I was looking for when I came here, as she used to eat it on rice.....did anyone else have a remedy like that?