My recipe. It is teh awesomes.
BETTER INSTRUCTIONS: Butter and a dash of mustard first. Mix it about 'til it's not lumpy. Then mix in the milk, salt and pepper, paprika and (optionally) onions or bay leaf. After that's mixed up properly, gradually stir in the cheese. Let it simmer and reduce for a bit until it's as thick as you want it to be, being careful not to burn the cheese*.
*it is possible to make this sauce very thick just by reducing it, so any thickeners such as flour or eggs are wildly unnecessary and just hurt the texture. Put it on a low medium and stir it every few minutes and you can basically keep going until it's the texture of chewed gum. Should you for some reason want to do that.
My WIP notes and quantities:
2:1 ratio of american cheese to milk (little easy on the cheese - technically more like1¾:1)
Mustard (1 tbsp/x)
Flour (3 tbsp/x = 1.5 tsp)
Butter (3 tbsp/x)
1 egg
Salt + pepper
a dash of paprika
Extras: onion, bay leaf.
Everything other than the cheese (butter + flour + mustard 'til free of lumps) goes first. Stir in everything but cheese. Then stir in egg, then the cheese.