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Melt the butter over medium heat in a skillet.
Whisk in the flour and cook for 1–2 minutes to form a roux.
Gradually pour in the milk, whisking constantly to prevent lumps.
Cook the mixture for 3–4 minutes, or until slightly thickened.
Stir in the cheddar cheese, Monterey Jack cheese, green chiles, cayenne pepper, salt, and pepper until the cheese is fully melted and the sauce is smooth.
Return the cooked chicken breasts to the skillet and spoon the cheese sauce over them.
Let the chicken simmer in the sauce for 2–3 minutes to soak up the flavors.
Garnish with chopped cilantro or sliced jalapeños, if desired.
Serve hot with rice, beans, tortillas, or a fresh side salad.