Combine mayonnaise, yogurt, basil, and lemon juice in bowl and season with salt and pepper to taste; cover and refrigerate until ready to serve.
Break ground turkey into small pieces in large bowl. Add zucchini, carrot, Parmesan, and pepper and gently knead with hands until well combined. Divide turkey mixture into 4 equal portions, then gently shape each portion into ¾‑inch-thick patty. Using your fingertips, press center of each patty down until about ½ inch thick, creating slight divot.
Season patties with salt. Heat oil in 12‑inch nonstick skillet over medium heat until just smoking. Transfer patties to skillet, divot side up, and cook until well browned on first side, 4 to 6 minutes. Flip patties, reduce heat to medium-low, and continue to cook until browned on second side and meat registers 160 degrees, 5 to 7 minutes. Transfer burgers to platter and let rest for 5 minutes. Serve burgers on buns, dolloped with mayonnaise mixture.
