Put the turkey, pepperoni, rosemary, half of the garlic and 1/2 teaspoon each salt and pepper in a large bowl and mix with your hands until combined. Form into four 1/2-inch-thick patties.
Heat a large cast-iron skillet over medium-high heat. Add the patties and cook, undisturbed, until browned, 5 minutes. Turn and cook until golden brown on the bottom, 3 more minutes. Meanwhile, whisk the mayonnaise, parsley, lemon zest and juice, and the remaining garlic in a small bowl. Season with salt and pepper. Top each patty with 2 slices cheese, cover and cook until the cheese melts, about 1 minute. Transfer the patties to a plate.
Toast the buns cut-side down in the skillet, then remove from the pan and spread with the herb mayonnaise. Season the tomato slices with salt and pepper. Sandwich the lettuce, burger patties and tomatoes on the buns.