Preheat oven to 350°F. Grease a 9x13-inch baking dish.
In a medium saucepan, heat oil over medium heat. Add onion and garlic; cook until softened, about 5 minutes. Stir in flour until combined. Add green chiles, chicken broth, cumin, chili powder, salt and pepper. Simmer for 10 minutes, stirring occasionally.
Pour 1/2 cup of the sauce into the prepared baking dish. Spread 1/4 cup of the sauce over each tortilla. Place 1/4 cup of the chicken in the center of each tortilla and sprinkle with cheese. Roll up and place in the baking dish.
Pour the remaining sauce over the top of the enchiladas. Sprinkle with remaining cheese. Bake for 25 minutes, or until cheese is melted and bubbly.
Interesting Facts
Enchiladas originated in Mexico and are now a popular Mexican-American dish.
Green chiles are a popular ingredient in Mexican cuisine and can be found in many dishes.
Monterey Jack cheese is a mild, semi-hard cheese that melts well and adds flavor to many dishes.