Heat a 10–12-inch skillet over medium-low heat, add 1tablespoonolive oil, then add the 1red onionsliced. Sauté, stirring about once a minute and watching closely, until the onions are glossy, soft, and browned, about 15 minutes. Transfer the caramelized onions to a bowl and set aside.
In a medium bowl, combine 1poundground turkey breast, 1egg, 2tablespoonchopped parsley, 1/2teaspooncurry powder, 1teaspoongarlic powder, 1/2teaspoonWorcestershire Sauce, 1/2teaspoonsalt, and 1/4teaspoonblack pepper. Mix gently until just combined.
Divide the mixture and shape into 4 even patties (one per hamburger bun). Handle the meat as little as possible to keep the patties tender.
Cook the patties on a preheated grill or in the same skillet (leave the pan oil in place) over medium heat. Cook until the centers are cooked through and no longer pink, about 5–6 minutes per side depending on thickness, or until an instant-read thermometer inserted into the center reads 165°F. Transfer cooked patties to a plate and let rest 2 minutes.
Place each burger patty on the bottom half of a hamburger bun, top each with a portion of the caramelized red onions, and cover with the top bun. Serve immediately.