Preheat oven to 400 degrees F.
Filling:
In a 10 1/2-inch cast iron skillet, heat the olive oil over medium heat. Add 1 cup onion, garlic, green pepper, cumin, and chili powder, and cook for about 3 to 5 minutes, or until the vegetables are tender. Add the lime juice, cannellini beans, chicken, broth, and green chiles, and stir until just combined. Remove from the heat and set aside.
Crust:
In a medium mixing bowl, beat the egg. Add the milk, butter, and cornbread mix, and blend well. Stir in 1/4 cup onion and the cheese. Pour over the chicken chili in the skillet. Bake for 25 to 30 minutes, or until the cornbread is golden brown. Remove from the oven and let cool for about 5 minutes.
If desired, top with sour cream, salsa, and cilantro before serving.
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