Combine mayonnaise and next 5 ingredients in a food processor or blender. Pulse 4-5 times or until smooth and well blended. Scrape down sides once or twice if necessary. Transfer to a small bowl and set aside.
Lay leave lettuce leaves on a flat surface. Place 1 turkey slice over the center of each leaf. Top each with two slices of mild cheddar and two slices of swiss cheese. Place two tomato slices over cheese. Spoon reserved basil sauce over tomato. Roll up lettuce like a burrito.