In a small bowl, combine the mayonnaise, Parmesan, lemon juice, mustard, ⅔ of the garlic, anchovy (if using), and ⅛ teaspoon black pepper.
Meanwhile, heat grill to medium-high. In a medium bowl, combine the turkey, Gruyère, and remaining 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.
Place the burgers on the buns and top with the Caesar spread.