Combine first 5 ingredients. Divide turkey mixture into 4 equal portions, shaping each into a 1/2-inch patty.
Place patties on a grill rack coated with cooking spray; grill 8 minutes on each side or until a thermometer registers 165°.
Place 1 lettuce leaf, if desired, on bottom half of each bun; top each with 1 burger, and 1 tomato slice, if desired. Spread 1 tablespoon Basil Aïoli on inside of each bun top. Place each top on 1 burger.