Melt butter in a large skillet over medium heat; cook and stir onion until tender and translucent, about 5 minutes. Mix in cream of chicken soup, sour cream, and cumin.
Grease a 9x12-inch microwave-safe baking dish and place 4 tortillas in a layer into the bottom of the dish. Top with 1 can green chilies, half the chicken, half the soup mix, half the Cheddar cheese, and half the Monterey Jack cheese. Repeat layers, ending with Monterey Jack cheese.
Cook in microwave oven on high until the casserole is bubbling and the cheese topping is melted, about 10 minutes.