Heat grill to medium-high. In a medium bowl, combine the turkey, Gruyère, and garlic. (Do not overwork the meat.)
Form the mixture into four ¾-inch-thick patties and season with the chili powder, ½ teaspoon salt, and ¼ teaspoon pepper.
Grill the patties, turning once (do not press), until an instant-read thermometer inserted in the center registers 160° F, 6 to 8 minutes per side.
Meanwhile, brush both sides of the onion rounds with oil and sprinkle with Cajun mix. Grill until blackened and tender, 5 to 6 minutes per side.
Place the burgers on the buns and top with the onions.