Heat olive oil in a large skillet over medium heat. Add chopped onion and cook until softened, about 5 minutes. Add minced garlic and cook for 1 minute more until fragrant.
Stir in the undrained diced green chiles, undrained diced tomatoes and green chiles, rinsed black beans, cumin, salt, and pepper. Bring to a simmer and cook for 5 minutes, stirring occasionally.
Remove the skillet from heat. Stir in the shredded chicken, sour cream, 1/2 cup Monterey Jack cheese, and 1/2 cup cheddar cheese. Mix well to combine.
Warm the flour tortillas slightly to make them pliable. You can do this by wrapping them in a damp paper towel and microwaving for 30 seconds, or by warming them one by one in a dry skillet.
Spread about 1/4 cup of the enchilada sauce in the bottom of a 9×13 inch baking dish.
Spoon about 1/2 cup of the chicken mixture into the center of each tortilla. Roll up the tortillas tightly and place them seam-side down in the prepared baking dish.
Pour the remaining enchilada sauce over the rolled enchiladas.
Top evenly with the remaining 1/2 cup Monterey Jack cheese and 1/2 cup cheddar cheese.
Bake for 20-25 minutes, or until the cheese is melted and bubbly and the enchiladas are heated through.
Let stand for 5 minutes before serving.
Notes
For a spicier enchilada, use hot green chiles or add a pinch of cayenne pepper to the chicken mixture.
You can assemble these enchiladas ahead of time and refrigerate them for up to 24 hours before baking. Add a few extra minutes to the baking time if baking from cold.
These enchiladas freeze well. Assemble and bake as directed, let cool completely, then cover tightly and freeze. Reheat from frozen at 350°F (175°C) until heated through, about 45-60 minutes.