Preheat a grill to medium high. Gently mix the turkey, 1 tablespoon mayonnaise, 2 teaspoons Worcestershire sauce, 2 teaspoons mustard, the red onion, lemon zest, 3/4 teaspoon salt and a few grinds of pepper in a large bowl. Form into four 1/2-inch-thick patties and place on a plate. Freeze until slightly firmer, 10 to 15 minutes.
Meanwhile, combine the lemon juice and grated garlic in a small bowl; let sit 5 minutes. Whisk in the remaining 1/3 cup mayonnaise, 1 teaspoon Worcestershire sauce, 1 teaspoon mustard, the grated parmesan, a pinch of salt and 1/4 teaspoon pepper.
Lightly oil the grill grates. Grill the turkey patties until cooked through but still juicy, about 4 minutes per side. Toast the buns on the grill, about 1 minute.
Toss 3 tablespoons of the dressing with the romaine and shaved parmesan; season with salt and pepper. Serve the burgers on the buns and top with mustard, the salad and the remaining dressing. Serve with watermelon and potato chips.