In a mixing bowl, combine the ground turkey with 3 tablespoons pesto. Mix gently, until just combined. Shape the mixture into four burgers.
Grill for 4-5 minutes per side until the internal temperature is 165 degrees. During the last minute of cooking, top each burger with a slice of cheese. Toast the burger buns on the grill, if desired.
Meanwhile, in a small bowl, stir together the mayonnaise with the remaining tablespoon of pesto.
Spread the top and bottom of each bun with the pesto mayonnaise. Top with a burger, tomato slice, and fresh basil. Season the tomatoes with flaky sea salt if desired.