Heat the oil in a large nonstick skillet over medium-high heat, add half of the onion and the garlic and cook, stirring, until soft and just begins to brown, about 4 minutes. Add the tomato paste, chili powder, 1/2 teaspoon salt and the cayenne pepper. Cook, stirring, until the oil is brick red, about 2 minutes. Add the chicken broth and bring to a simmer. Cook until thickened slightly, about 3 minutes. Add the shredded chicken and stir until just warmed through, remove from heat and cool to room temperature.
Whisk together the sour cream, cilantro, lime juice and zest in a small bowl. Toss together the corn and pepper in another small bowl.
Layer the dip in a medium, straight-sided 3-quart glass trifle bowl or your favorite serving dish. Spread the cornbread on the bottom, then top, in even layers, with the tomatoes, beans, cheese, cilantro-lime sour cream, chicken chili, the remaining chopped onions and the corn-pepper mixture.
Wrap and refrigerate until chilled, at least 1 hour or overnight. Serve with chips for dipping.