Step 1: Preheat your oven to 375°F (190°C). Lightly grease a 9x13 inch baking dish.
Step 2: In a large skillet, heat olive oil over medium heat. Add onion and cook until softened, about 5-7 minutes. Add minced garlic and cook for another minute until fragrant.
Step 3: Stir in the diced green chiles (undrained), green enchilada sauce, and chicken broth. Bring to a simmer, then reduce heat and let it gently cook for 5 minutes. Remove from heat and stir in the sour cream and fresh cilantro.
Step 4: In a large bowl, combine the shredded cooked chicken, 1 cup of Monterey Jack cheese, 1/2 cup of cheddar cheese, and 1/2 cup of the prepared green chile sauce. Mix well and season with salt and pepper to taste.
Step 5: To assemble the enchiladas, lightly warm the corn tortillas one by one (either in the microwave for 15-20 seconds or quickly dipped in warm oil) to make them pliable. Dip each tortilla briefly into the remaining green chile sauce.
Step 6: Place about 1/4 cup of the chicken mixture down the center of each tortilla. Roll up tightly and place seam-side down in the prepared baking dish.
Step 7: Pour the remaining green chile sauce evenly over the rolled enchiladas. Sprinkle with the remaining 1 cup of Monterey Jack cheese and 1/2 cup of cheddar cheese.
Step 8: Bake for 20-25 minutes, or until the enchiladas are heated through and the cheese is bubbly and lightly golden brown.
Step 9: Remove from oven and let stand for 5-10 minutes before serving. Garnish with additional sour cream, avocado, or cilantro if desired.