Preheat the oven to 375°F (190°C) and lightly grease a 9x13-inch baking dish.
In a large skillet, melt 2 tablespoons butter over medium heat. Add the chopped onion and sauté until soft, about 4–5 minutes.
Add the minced garlic and cook 30 seconds, then stir in 1 can (14 oz) green chiles, cumin, chili powder, 1 teaspoon salt, and 1/2 teaspoon black pepper; cook 1–2 minutes.
Stir in the shredded chicken until evenly coated, then remove from heat and stir in 1 cup grated Colby Jack cheese until melted; set the filling aside.
In a separate skillet, melt 3 tablespoons butter over medium heat. Whisk in 3 tablespoons flour and cook 1 minute to make a roux.
Slowly whisk in the warmed 2 cups chicken broth until smooth and slightly thickened, then reduce heat to low.
Stir in 1 cup sour cream, the second can (14 oz) green chiles, 1 teaspoon onion powder, 1 teaspoon garlic powder, 1/2 teaspoon salt, and 1/2 teaspoon black pepper until smooth and lump-free; remove from heat.
Wrap tortillas in a clean towel or paper towels and microwave on HIGH for 30 seconds to 1 minute until warm and pliable.
Spoon a thin layer of the sauce over the bottom of the prepared baking dish.
Place a tortilla on a work surface, spoon about 1/4 cup chicken filling down the center, roll tightly, and place seam-side down in the baking dish. Repeat with remaining tortillas.
Ladle the remaining sauce evenly over the enchiladas and sprinkle 1 1/2 cups shredded Colby Jack cheese over the top.
Bake in the preheated oven for 20–25 minutes, until bubbly and cheese is lightly browned. Let rest 5–10 minutes, then garnish with chopped cilantro and serve.