Preheat the oven to 350°F. Grease a 9×13-inch baking pan and set aside.

Make the enchilada sauce: In a small saucepan over medium heat, melt 4 tablespoons unsalted butter. Add 1 tablespoon minced onion, 1 minced garlic clove, 1/4 teaspoon kosher salt, and 1 teaspoon ground cumin; cook about 1 minute, until the onion is softened. Stir in 5 tablespoons all-purpose flour and cook about 30 seconds.

Gradually whisk in low-sodium chicken broth, starting with 2 cups and adding up to the remaining 1 cup (2–3 cups total) as needed, until the mixture is smooth and sauce-like. If you add too much liquid, simmer briefly to reduce. Remove from heat and stir in 1 cup sour cream and 4 ounces green chiles. Set the sauce aside to cool slightly.

Reserve 1 cup of the prepared sauce for the filling, and pour 1/2 cup of the prepared sauce into the bottom of the greased baking pan.

Make the filling: In a medium bowl, combine 2½ cups shredded cooked chicken, 15 ounces drained black beans, 4 chopped green onions, 1 tablespoon lime juice, 1 cup of the Monterey Jack cheese, and the reserved 1 cup of sauce. Stir until evenly combined.

Assemble the enchiladas: Spoon about 1/2 cup of the filling onto each of the 8 large flour tortillas, roll each tortilla up, and place seam-side down in the prepared baking pan.

Pour the remaining prepared sauce evenly over the rolled enchiladas. Sprinkle the remaining 1 cup Monterey Jack cheese evenly over the top.

Bake uncovered in the preheated oven for 20–30 minutes, until the cheese is melted and the edges of the tortillas are beginning to brown and crisp.

Remove from the oven and, if desired, top with chopped fresh cilantro, chopped onion, chopped jalapeño peppers, avocado slices, and cotija cheese before serving.