Melt butter in a saucepan over medium heat. Add the chopped onion and cook until soft. Add garlic and cook for another minute. Stir often to prevent the garlic from burning.

Sprinkle flour over the cooked onion mixture and stir well. Slowly add chicken broth while whisking. Continue cooking until the mixture becomes smooth and slightly thickened.

Reduce the heat to low before adding sour cream. Stir gently until completely combined. Add green chilies, cumin, chili powder, salt, and black pepper for extra flavor.

Pour about half a cup of sauce into the shredded chicken. Add half the Monterey Jack cheese and chopped cilantro. Stir everything gently until the filling becomes creamy and well mixed.

Preheat your oven to 375°F. Lightly grease a 9x13-inch baking dish with vegetable oil. Spread a small amount of sauce across the bottom to keep the tortillas moist.

Warm each tortilla briefly so it becomes flexible. Add several spoonfuls of chicken filling down the center. Roll each tortilla tightly and place it seam-side down inside the baking dish.

Pour the remaining creamy sauce evenly over the rolled tortillas. Sprinkle the remaining Monterey Jack and cheddar cheese across the top. Make sure most of the tortillas are covered.

Cover the dish loosely with foil and bake for 20 minutes. Remove the foil and continue baking for another 10 minutes. The cheese should be melted, bubbly, and lightly golden.

Remove the enchiladas from the oven and let them rest for five minutes. Sprinkle fresh cilantro over the top. Serve warm while the cheese is still soft and delicious.