Prepare the Mexican Chicken according to its recipe directions. Once cooked, drain away any excess liquid from the chicken.
Preheat the oven to 400 degrees F. Cover a baking sheet with foil and position a baking rack over the sheet.
Assemble the burritos by placing a heaping ½ cup filling onto each tortilla. If using raw tortillas, leave them uncooked at this stage. Add the desired amount of cheese, then fold and roll each tortilla into a burrito.
Arrange the burritos on the baking rack. Lightly brush both sides with olive oil or coat them with nonstick cooking spray.
Bake for 18-20 minutes, or until the tortillas are golden. Broil until they reach your preferred level of crispness, then turn the burritos over and broil the opposite side until golden.
While the burritos bake, make the sauce. Add the butter and olive oil to a medium saucepan and melt over medium heat.
Whisk in the flour and cook for 3 minutes, stirring continuously.
Turn the heat to low. Slowly whisk in the warmed chicken broth, then add the cumin, salt, and pepper. Bring the mixture to a simmer while stirring and cook until thickened, approximately 2-3 minutes.
Take the saucepan off the heat. Stir in the green chilies and cheese, continuing until the cheese has melted. Stir in the sour cream. Add hot sauce to taste, if desired.
Serve the baked burritos topped with the Cheesy Green Chili Sour Cream Sauce and your choice of optional toppings.
Lightly spray the burritos with olive oil. Working in batches, arrange them seam side down in the air fryer basket without allowing them to touch.
Air fry at 375 degrees F for approximately 5-6 minutes, or until the tops are golden and crisp. Flip the burritos and continue air frying for another 4-5 minutes, or until the second side is golden and crispy.