Preheat oven to 400°F and set a 9x13-inch casserole dish nearby.
Cut each chicken breast in half lengthwise (you should have 4 small or about 6 pieces) and season both sides with 1 tsp salt and 1/4 tsp black pepper.
Heat a large nonstick or well-seasoned skillet over medium heat until hot.
Add chicken to the skillet in a single layer and cook about 3 minutes per side, or until golden and the thickest part registers 165°F on an instant-read thermometer. Cook in 2 batches if necessary to avoid overcrowding.
Arrange the cooked chicken pieces flat in the 9x13 casserole dish so they do not overlap.
In a medium bowl, stir together 2/3 of the shredded Mexican cheese, 1 cup sour cream, and the contents (with their juice) of the two 7-oz cans (14 oz) diced green chiles until evenly combined.
Pour the sour cream–cheese–chile mixture evenly over the chicken, then sprinkle the remaining 1/3 of the shredded Mexican cheese on top.
Bake uncovered at 400°F for about 15 minutes, or until the cheese is melted and the sauce is bubbling at the edges.
Remove from the oven, garnish with the diced avocado, diced roma tomato, and chopped cilantro, and serve.