Begin by heating the olive oil in a skillet over medium heat. Add the sliced red onions and a pinch of salt. Sauté for about 15-20 minutes, stirring occasionally until the onions are golden brown and caramelized. Remove them from the skillet and set aside.
In a mixing bowl, combine the ground turkey breast, egg, chopped parsley, curry powder, garlic powder, Worcestershire sauce, salt, and black pepper. Mix gently until just combined, being careful not to overmix.
Divide the turkey mixture into four equal portions and shape each portion into a patty, about 3/4 inch thick. Make a slight indentation in the center of each patty to help them cook evenly.
In the same skillet used for the onions, add a little more olive oil if necessary and heat over medium-high heat. Add the turkey patties and cook for about 5-6 minutes on each side, or until the internal temperature reaches 165°F (74°C).
Toast the hamburger buns lightly on the skillet or grill if desired. Place the cooked turkey patties on the bottom half of each bun, top generously with the caramelized onions, and finish with any additional toppings you love, such as lettuce or tomato.
Top with the other half of the bun, serve with your favorite sides, and enjoy these mouthwatering Turkey Burgers with Caramelized Red Onions!