Preheat the Oven:
Preheat your oven to 375°F (190°C) and lightly grease a 9×13-inch baking dish with a little olive oil or cooking spray.

Prepare the Filling:
Heat the olive oil in a large skillet over medium heat. Add the chopped onion and cook for 3-4 minutes, or until softened. Add the minced garlic and cook for another 1 minute until fragrant. Stir in the shredded rotisserie chicken, enchilada sauce, lime juice, salt, and black pepper. Cook for 5 minutes, stirring occasionally, until everything is heated through.

Add Beans and Rice:
Stir in the black beans and cooked rice, mixing until everything is well combined and the mixture is evenly coated with the sauce. Taste and adjust seasoning if needed. Remove the skillet from the heat.

Assemble the Burritos:
Lay each tortilla flat on a clean surface. Divide the chicken mixture evenly among the tortillas, placing it in the center of each one. Sprinkle about 2 tablespoons of the cheddar and Monterey Jack cheese on top of the filling. Carefully roll up each tortilla, folding in the sides as you go to form a burrito.

Arrange in the Baking Dish:
Place the rolled burritos seam-side down in the prepared baking dish. Once all the burritos are in the dish, sprinkle the remaining cheddar and Monterey Jack cheese on top.

Bake the Burritos:
Bake the burritos in the preheated oven for 15-20 minutes, or until the cheese is melted and bubbly, and the tortillas are slightly crispy around the edges.

Serve and Enjoy:
Once the burritos are done, remove them from the oven and allow them to cool for a few minutes. Serve with a dollop of sour cream, a drizzle of hot sauce, and a sprinkle of chopped cilantro, if desired.