Preheat the oven to 400° F and line a rimmed baking sheet with parchment paper or a silicone baking mat, and set aside.
Heat a large skillet over medium heat and add the oil, garlic, bell peppers, and onions, cooking until they are very soft, 8-10 minutes.
Reduce the heat to low, add the green chilies with their juice, and cook for an additional 2 minutes. Then remove the skillet from the heat.
While the skillet is cooling, prepare the chicken to be stuffed. Make 4-6 diagonal slits in each chicken breast, being careful to not slice all the way through.
Salt and pepper both sides of the chicken and fill each slit evenly with the vegetable mixture.
Season the tops of the stuffed chicken with cumin and chili powder, and top each chicken breast with 2 tablespoons of cheese.
Bake until done, 20-22 minutes.
Allow to rest for 5 minutes, then garnish each chicken breast with optional sour cream and cilantro as desired, and serve with a lime wedge to squeeze over the chicken.