Preheat Oven
Preheat your oven to 375°F (190°C). Lightly grease a 9x13-inch baking dish.
Prepare the Filling
In a large bowl, mix shredded chicken, 1 cup of Monterey Jack cheese, and ½ cup of cheddar cheese.
Assemble the Enchiladas
Spoon about 2-3 tablespoons of the chicken mixture onto each tortilla.
Roll up the tortillas tightly and place them seam-side down in the prepared baking dish.
Make the Sauce
In a medium saucepan, melt the butter over medium heat.
Whisk in the flour and cook for 1-2 minutes to create a roux.
Slowly pour in the chicken broth, whisking constantly to prevent lumps. Cook until the sauce thickens slightly.
Stir in the sour cream and green chiles until fully combined. Remove from heat.
Pour and Top
Pour the creamy sauce evenly over the enchiladas in the baking dish.
Sprinkle the remaining Monterey Jack and cheddar cheese on top.
Bake
Bake uncovered for 20-25 minutes, or until the cheese is melted and bubbly.
If desired, broil for an additional 2-3 minutes to lightly brown the cheese.
Serve
Let the enchiladas rest for 5 minutes before serving. Garnish with chopped cilantro or green onions if desired.