I make potato, cheese, and bacon soup often and when I want to thicken the milk, I put about two tablespoons of flour into two tablespoons of melted butter in a pot and cook a couple minutes to get the edge off the flour. I believe this is a roux?
Anyway, once I've done this, I add cold milk to the butter and flour. Chef John always says a hot roux with cold milk will keep you from having lumps.
I don't have the heat on real high, but I do stir the milk for about 10 minutes and you can tell when it starts getting thicker. When it's thick enough for me (or when I get tired of stirring), I remove the pot from the stove and then stir in the cheese. I don't let my milk get real thick because the cheese will thicken it up as well. When the cheese is all melted, I add the whole thing to the potatoes and bacon.
Recently, I've been using Parmesan cheese, cream, and Alfredo sauce in my potatoes and bacon in the slow cooker, and then stirring the cheese in when the potatoes are tender. I'm still experimenting with that, though.