Place oil in 12-inch nonstick skillet and set aside. Combine water and baking soda in small bowl. Place turkey in large bowl. Using your hands, break up meat into rough ½-inch pieces. Drizzle baking soda mixture evenly over turkey, followed by soy sauce and melted butter. Evenly sprinkle panko, Parmesan, gelatin, pepper, and salt over turkey mixture. Using your hands, gently toss to combine.
Divide meat into 4 lightly packed portions, about 4 ounces each. Gently flatten 1 portion into patty about ½ inch thick and about 4 inches in diameter. Transfer patty directly to prepared skillet and repeat with remaining portions.
Heat skillet over medium heat. When patties start to sizzle, cover skillet and cook until patties are well-browned on bottom, about 2 ½ minutes (if patties are not browned after 2 ½ minutes, increase heat). Carefully flip patties over and continue to cook until second side is well browned and burgers register 160 degrees, 2 ½ to 3 minutes longer. If using cheese, place 1 slice on each burger about 1 minute before burgers finish cooking. Transfer burgers to plate and let rest for 5 minutes, then transfer to buns and serve.
