1. Place oil in a 10- to 12-inch frying pan over medium-high heat. Crumble turkey into pan. Add onions. Cook and stir until onions are limp, about 3 minutes. Add chili powder and cumin and stir 2 to 3 minutes. Add 3/4 cup water, spaghetti sauce, and beans. Bring to a boil. Reduce heat and simmer, uncovered, 10 to 12 minutes, stirring occasionally.
2. Meanwhile, in a bowl, lightly beat together egg and milk. Stir in chilies and cornbread mix.
3. Pour hot turkey mixture into a shallow 2 1/2- to 3-quart baking dish. Sprinkle cheese over turkey mixture. Spread cornbread batter evenly over cheese. Bake, uncovered, in a 375° oven until cornbread topping is golden brown, 25 to 30 minutes. Scoop out servings with a spoon.