Heat 1 tablespoon olive oil in a large skillet over medium-low heat. Add 1 red onion (sliced) and sauté about 15 minutes, stirring about once a minute, until the onion is glossy, soft, and browned. Watch constantly to prevent burning. Transfer the caramelized onions to a plate and set aside.
In a bowl, combine 1 pound ground turkey breast, 1 egg, 2 tablespoons chopped parsley, 1/2 teaspoon curry powder, 1 teaspoon garlic powder, 1/2 teaspoon Worcestershire Sauce, 1/2 teaspoon salt, and 1/4 teaspoon black pepper. Mix gently until just combined.
Divide the turkey mixture into 4 equal portions and shape each into a patty roughly the diameter of the hamburger buns.
Preheat a grill to medium-high or heat a clean nonstick skillet over medium-high heat. If using the same skillet as the onions, wipe out any burned bits but a thin film of the remaining oil is fine; do not add any ingredients not listed. Cook the patties 5–7 minutes per side, or until the internal temperature reaches 165°F (74°C) and no pink remains in the center.
Place each cooked patty on the bottom half of a hamburger bun (4 buns). Top each patty with some of the reserved caramelized red onions, then cover with the top bun. Serve immediately.