Spread the reheated chili evenly over the bottom of a serving bowl with a capacity of at least 8 cups. Place in a microwave or warm oven to keep hot.
Melt the butter over medium heat in a stainless steel everyday pan or 12-inch skillet. Add the minced onions and stir to coat the onions with butter. Saute until the onions are fragrant and translucent at the edges. Stir in the diced green chiles and raise the heat to high. Cook, stirring constantly, for 1 minute.
Sprinkle the chili powder and cumin over the top and stir for 30 seconds. Scatter the flour over the mixture and whisk to evenly distribute everything. Drop the heat to medium high. Cook for 2 minutes, stirring constantly.
Slowly whisk the milk into the onions and chiles until smooth. Let the sauce come to a simmer, whisking slowly but constantly, letting it bubble and become thick, about 3 minutes. Add the sliced American cheese strips, shredded Cheddar, and shredded Monterey Jack or Mexican blend all at once. Stir gently until the cheese is completely melted and has formed a smooth queso.
Pour over the warmed chili. Serve hot, garnished with cilantro leaves if you'd like, accompanied by copious amounts of tortilla chips and sturdy vegetables for dipping.