Grease an 8-inch baking dish with the butter and set it aside.
Heat the olive oil in a nonstick skillet over medium heat. Add the ground turkey, the onions, green peppers, and garlic and cook until the turkey is no longer pink and is cooked through, about 8 minutes. Drain off any excess fat and sprinkle the meat mixture with the cumin.
Add the beans, tomatoes, chiles and adobo sauce to the skillet and bring to a boil over high heat. Reduce the heat to medium and simmer until heated through and slightly thickened, about 5 minutes. Remove pan from the heat and stir in the cheese and cilantro.
Spread the turkey mixture in the prepared baking dish, pressing down on it with the back of a spoon to make an even, compact layer.
Combine the cornbread mix with milk and egg. Spread the cornbread batter over the turkey mixture and bake until the cornbread is golden-brown, 20 to 25 minutes. Let the tamale pie stand for 5 minutes before cutting into squares and serving.