Preheat the oven to 350°F and grease a 9×13-inch skillet or baking dish.

Combine the shredded chicken with 1 cup of the grated cheese in a bowl; toss until evenly mixed.

Warm each flour tortilla briefly so they bend easily, fill with about 1/3 cup chicken mixture, and roll tightly. Arrange them seam-side down in the greased pan.

Melt the butter in a skillet over medium heat; you want it foaming but not brown.

Stir in the flour and whisk for 1 minute until it smells nutty and starts to thicken slightly.

Add the chicken broth while whisking and cook over high heat, whisking until the sauce thickens and bubbles.

Remove from heat and stir in the sour cream and diced green chiles; the sauce will loosen and smell tangy and bright.

Pour the creamy sauce evenly over the rolled tortillas, then sprinkle the remaining 2 cups of cheese across the top.

Bake for 20–23 minutes until bubbling at the edges and the cheese melts.

Broil for 2–3 minutes to brown the cheese lightly; watch closely so it doesn’t burn.

Let rest 5 minutes before serving; the sauce will thicken and the enchiladas will set.