In a large bowl, combine the ground turkey, grated onion, mayonnaise, Worcestershire sauce, garlic, smoked paprika, onion powder, salt, and black pepper. The mayonnaise adds fat and moisture, which is critical for preventing dry turkey burgers.Mix everything gently until just incorporated, then divide the mixture into 4 equal portions and shape them into patties about 3/4-inch thick. Press a slight indent into the center of each patty with your thumb to keep them flat during cooking.
Preheat the air fryer to 375 degrees F for 3 minutes.
Place the turkey burger patties in the air fryer basket in a single layer, making sure they are not touching. Cook for 8 minutes.
Flip the burgers carefully using a thin spatula. Cook for another 4 minutes, then place a slice of cheddar cheese on each patty. Cook for 1 more minute until the cheese is melted and the internal temperature reaches 165 degrees F.
While the burgers finish cooking, lightly toast the brioche buns in the air fryer for 1 to 2 minutes or in a toaster.
Assemble the burgers on the toasted buns with lettuce, tomato, pickles, and your favorite condiments. Serve immediately while hot and juicy.