Preheat oven to 375°F (190°C) and lightly grease a 9x13-inch baking dish.
In a medium skillet over medium heat, melt butter. Add flour and whisk for about 1 minute until slightly golden.
Slowly whisk in chicken broth, simmer until sauce thickens, 2–3 minutes.
Remove from heat and stir in sour cream, green chiles, cumin, chili powder, salt and pepper. Adjust consistency with extra broth if needed.
Combine shredded chicken, half of the sauce, chopped onion, garlic, and 1/2 cup cheese in a bowl.
Spoon filling into each tortilla, roll tightly, and place seam-side down in the baking dish.
Pour remaining sauce over enchiladas, top with remaining cheese, cover with foil and bake 15 minutes. Remove foil and bake 5–8 minutes more until cheese is bubbly.
Let rest 5 minutes, then garnish with cilantro and lime wedges and serve warm.
Notes
Warm tortillas before rolling to prevent cracking.
Leftovers keep in the fridge for 3–4 days and reheat well in the oven.
Swap chicken for turkey or shredded pork if desired.