Prep and preheat: Heat the oven to 375°F (190°C). Lightly grease a 9x13-inch baking dish.
Sauté aromatics: Warm a skillet over medium heat with a drizzle of oil. Add the chopped onion and cook until soft and translucent, about 5 minutes.
Stir in the garlic and cook 30 seconds.
Make the roux: Push the onions to the side and melt the butter in the skillet. Sprinkle in the flour and whisk for 1 minute to form a paste.
Build the sauce: Slowly whisk in the chicken broth until smooth. Simmer 2 to 3 minutes, stirring, until slightly thickened.
Take off the heat and whisk in sour cream, cumin, chili powder, salt, and pepper. Stir in the diced green chiles. Taste and adjust seasoning.
Set aside some sauce: Pour about 1 cup of the sauce into the bottom of the baking dish and spread it out.
Keep the rest warm.
Warm tortillas: Briefly warm each tortilla in a dry skillet or microwave until pliable. This helps prevent cracking and makes rolling easier.
Assemble filling: In a bowl, combine the shredded chicken with 1 cup of the shredded cheese and 1/2 cup of the sauce. Mix to coat.
Roll the enchiladas: Spoon some filling down the center of a tortilla.
Roll snugly and place seam-side down in the baking dish. Repeat with the remaining tortillas and filling.
Sauce and cheese: Pour the remaining sauce evenly over the enchiladas, covering all edges so they don’t dry out. Top with the remaining cheese.
Bake: Cover loosely with foil and bake for 15 minutes.
Uncover and bake another 8 to 10 minutes, until the cheese is melted and bubbling and the edges are slightly golden.
Rest and garnish: Let the enchiladas rest 5 to 10 minutes to set. Sprinkle with chopped cilantro and serve with lime wedges.