Preheat the oven to 400°F (200°C). Spray a 9x13-inch glass or ceramic baking dish with olive oil or nonstick spray and set aside.
Trim any fat or undesirable parts from the chicken breasts.
Heat 1 tablespoon olive oil in a large nonstick skillet over medium-high heat.
Season the chicken breasts evenly with 1 teaspoon taco seasoning and salt and pepper to taste.
Sear the chicken breasts 3 minutes per side in the hot skillet; they will not be fully cooked. Transfer to a cutting board and let rest until cool enough to handle.
While the chicken cooks, drain the roasted green chiles in a colander. Split each chile in half, remove any seeds, and pat the chiles dry between paper towels to remove excess moisture.
Cut each rested chicken breast lengthwise into 3 or 4 strips.
Spoon 1/4 cup of the salsa verde into the bottom of the prepared baking dish and spread it into an even layer.
Arrange the chicken strips crosswise down the length of the baking dish on top of the sauce.
Cover the chicken strips with the drained green chile strips. Pour the remaining salsa verde over the chiles and sprinkle 1 cup grated Mexican cheese evenly on top.
Bake for 20–25 minutes, until the cheese is completely melted and the dish is heated through. If you want browned cheese, broil 1–2 minutes at the end, watching carefully to avoid overcooking the chicken.
Serve hot with optional sour cream, salsa, chopped avocado, or Green Tabasco Sauce at the table.