In a large skillet, heat the olive oil over medium heat. Add the sliced red onion and stir occasionally. Cook for about 15-20 minutes until golden brown and sweet. Lower heat if onions brown too quickly. Set aside when done.
In a mixing bowl, combine ground turkey breast, egg, chopped parsley, curry powder, garlic powder, Worcestershire sauce, salt, and black pepper. Mix gently until well combined, avoiding overmixing.
Divide the turkey mixture into four equal portions. Shape each portion into a patty with a slight indentation in the center to help cook evenly.
In the same skillet used for onions, increase heat to medium-high. Cook the turkey patties for 5-7 minutes on each side or until internal temperature reaches 165°F (75°C). Optionally, add a slice of cheese on top during the last minute to melt.
Toast the hamburger buns if desired. Assemble burgers by placing a turkey patty on the bottom bun and topping with caramelized onions. Add any preferred toppings such as lettuce, tomato, or avocado. Place top bun and serve.