In a large ovenproof skillet, heat the olive oil, onion, peppers, and garlic over medium-high heat for 3-5 minutes until soft and lightly browned.
Add the pasta, chicken, reserved pasta water, sour cream, salsa verde, canned chiles, and 1 cup of the cheese to the vegetable mixture. Stir together and taste; add salt, pepper, or more salsa verde to your preference. The mixture should be slightly soupy.
Top the skillet with the rest of the cheese. Broil (if the dish is already hot) or bake at 400F for 20-30 minutes until the dish is bubbly and the cheese on top has browned.
Serve hot with cilantro and lime wedges. Enjoy!