Prep. Preheat the oven to 375°F and lightly grease a 12 inch skillet or 9x13 dish.
Start roux for sauce. In a medium saucepan, melt the butter and whisk in the flour. Cook for 1 minute, then slowly whisk in the chicken broth until smooth and slightly thickened.
Finish sauce. Reduce heat to low and stir in the sour cream, green chiles, garlic powder, onion powder, salt, and pepper. Warm gently and remove from heat.
Mix filling. In a bowl, combine the shredded chicken with 1/2 cup of the sauce and 1/2 cup of the cheese.
Assemble. Fill each tortilla with the chicken mixture, roll, and place seam side down in the skillet. Pour remaining sauce over top and sprinkle with the rest of the cheese.
Bake. Bake for 20 to 25 minutes, until hot and bubbling. Let cool for 5 minutes before serving.