Heat broiler. In a large bowl, combine the turkey, zucchini, carrot, garlic, thyme, salt, pepper, and egg. Form the mixture into 4 patties.
Heat 1 tablespoon of the oil in a large skillet over medium heat. Cook the patties, turning once, until no pink remains, 4 to 5 minutes per side.
Meanwhile, place the bread on a baking sheet and brush with the remaining oil. Broil until golden brown and crisp, about 1½ minutes.
Transfer the bread to individual plates. Top with the lettuce and burgers. If desired, combine the mayonnaise and lemon juice and serve with the burgers.