In a medium bowl, combine turkey, onion, Italian seasoning, salt, and pepper. Mix gently with your hands. Divide mixture in half and form into burgers.
Heat a grill pan or nonstick skillet over medium heat. Add burgers and cook, turning once, until browned on both sides and internal temperature reaches 165℉, about 4 minutes per side.
Take off heat, top burgers with cheese, and cover with a lid to melt cheese.
Lay wraps on a countertop. Spread ½ tablespoon mayonnaise in the middle of each wrap. Top with lettuce, tomato, and turkey burger.