Combine turkey, egg, bread crumbs, garlic, onion powder, poultry seasoning, cumin, chili powder, salt, white pepper, and cayenne in a large mixing bowl and mix by hand. Shape into 4 patties. Refrigerate for 30 minutes to absorb all of the flavors.
Preheat an outdoor grill for medium-high heat and lightly oil the grate.
Place patties on the preheated grill, close the lid, and let them sit for 6 to 8 minutes. Flip and continue to cook until browned and no longer pink, 7 to 9 minutes more. An instant-read thermometer inserted into the center of a patty should read at least 165 degrees F (74 degrees C). Serve on buns immediately.