In a medium bowl, combine the turkey, zucchini, mushrooms, salt, pepper, ketchup, mustard, and 1 tablespoon of water. Form into six patties. Brush both the patties and the sliced onion with vegetable oil. Grill both, flipping once, until the burgers are cooked through and the onions are starting to brown, about 8 minutes. In the last few minutes of cooking, place the English muffins or buns on the grill to tast.