Mix the turkey, bread crumbs, onion, eggs, applesauce, chili powder, paprika, and kosher salt in a large bowl until thoroughly combined. Divide mixture into 8 equal portions, then shape into patties.
Heat olive oil in a large skillet over medium-high heat. Arrange the patties in the skillet and cook until browned, 3 to 5 minutes on each side. Place the browned patties on a paper towel-lined plate to drain. Transfer the patties to a 9×13-inch baking dish.
Bake the turkey burgers in the preheated oven until no longer pink in the center and the juices run clear, about 10 minutes more. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Serve the turkey burgers on the hamburger buns.