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 trace of 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 1 1/2 minutes. Transfer the bread to individual plates. Top with the lettuce leaves and burgers. If using, combine the mayonnaise and lemon juice in a small bowl and serve with the burgers.
Upgrade: For extra zest, rub the bread with the cut side of a peeled, halved garlic clove immediately after removing the bread from the oven.