Preheat an outdoor grill for medium-high heat and lightly oil the grate.
Mix turkey, 1/4 cup barbeque sauce, green onion, oil, chili powder, and cumin together in a bowl; shape into 4 patties.
Grill patties on the preheated grill until cooked through, 4 to 5 minutes per side. An instant-read thermometer inserted into the center should read 165 degrees F (74 degrees C). Serve each burger on a bun and smother each burger with remaining barbeque sauce.