In a large skillet over medium heat, warm the olive oil. Add the chopped onion and cook until soft, about 5 minutes. Mix in the minced garlic and cook until fragrant, roughly 30 seconds.

To the skillet, add the shredded chicken and half a cup of enchilada sauce, stirring until the chicken is well coated. Add the lime juice, and season with salt and pepper to your liking.

Assemble the burritos by placing a tortilla on a flat surface. In the center of each tortilla, add a scoop of rice and black beans. Spoon the chicken mixture on top, then sprinkle with a portion of both cheddar and Monterey Jack cheeses, reserving about half a cup of cheese for topping.

Roll each tortilla tightly to enclose the filling and place seam-side down in a baking dish.

Drizzle the remaining enchilada sauce over the rolled burritos and sprinkle with the reserved cheese.

Bake in the preheated oven until the cheese is melted and bubbly, about 15 minutes.

Garnish with chopped cilantro, and serve with sour cream and hot sauce on the side if desired.