Pat the chicken breasts dry and place them on a cutting board.
Use a sharp knife to cut a pocket into the side of each chicken breast, being careful not to cut through.
Brush both sides of each chicken breast with the olive oil.
In a small bowl, combine paprika, 1/2 teaspoon salt, garlic powder, dried basil, and onion powder.
Sprinkle the spice mixture evenly over both sides of the chicken and rub it in with your hands.
In a medium bowl, mix together the softened cream cheese, grated Parmesan, grated mozzarella, chopped spinach, minced garlic, red pepper flakes, and the remaining 1/2 teaspoon salt until well combined.
Spoon the cheese-and-spinach mixture into the pocket of each chicken breast and press the opening closed to keep the filling inside.
Place the stuffed chicken breasts in a single layer in a greased air fryer basket; cook in batches if needed to avoid overlap.
Air fry at 375°F (190°C) for 15–20 minutes, or until the chicken is browned and an instant-read thermometer in the thickest part reads 165°F (74°C).
If cooking multiple batches, keep finished chicken covered loosely with foil while you cook the remaining batches.