Heat a grill to medium (350° to 450°). Mix turkey, zucchini, mushrooms, salt, pepper, ketchup, mustard, and 1 tbsp. water in a medium bowl. Form into 6 patties.
Brush patties and onion rings with oil. Grill both, turning once, until burgers are cooked through, about 8 minutes, and onion rings are starting to brown, about 3 minutes. In last few minutes, toast muffins on grill.
Assemble burgers on muffins with lettuce, onion, and any additional toppings you like.