In a large bowl, combine the chicken, orange juice, lime juice, garlic, cumin, and drained green chiles. Mix well to incorporate.
On a work surface, working with one at a time, place one quarter of the chicken mixture (almost 3/4 cup) onto the bottom third of a tortilla. Sprinkle each with 2 tablespoons cheese. Lift the edge nearest you and wrap it around the filling. Fold the left and right sides in toward the center and continue to roll into a tight cylinder. Set aside, seam side down, and repeat with remaining tortillas and filling.
Preheat air fryer to 400 degrees F.
Lightly spray all sides of the chimichangas with oil. Place 2 of the chimichangas seam side down in the air fryer basket. Make sure each chimichanga is tightly wrapped and seam side down in the basket or it will open during cooking. Cook for 7 to 8 minutes, flipping halfway, until golden and crisp. For a toaster oven-style air fryer, cook at 350 degrees F, for the same amount of time. Repeat with the remaining chimichangas.
To serve, place ¾ cup shredded lettuce on each plate. Place a chimichanga on top, along with 2 tablespoons pico de gallo, 1 tablespoon of sour cream or plain Greek yogurt, and avocado. Garnish with cilantro. Serve immediately.