Heat grill to medium-high. Once it is hot, clean the grill grate with a wire brush. Just before grilling, oil the grill grate.
Mix together the avocado, lime juice, and ¼ teaspoon each salt and pepper in a small bowl.
Gently mix together the turkey, Cheddar, grated onion, ¾ teaspoon salt, and ¼ teaspoon pepper with your hands in a medium bowl until just combined. Form the turkey mixture into four ¾-inch-thick patties. Use your fingers to make a shallow well in the top of each patty (this will prevent overplumping during cooking).
Grill the burgers until cooked through and an instant-read thermometer inserted in the center registers 165° F, 6 to 8 minutes per side.* Grill the buns, split-side down, until toasted, 5 to 10 seconds per side.
Stack the burgers, sliced onion, and avocado mixture between the buns. *This recipe includes the Real Simple test kitchen’s preferred cooking temperatures (considered safe by many experts) for burgers cooked to their juicy best. For maximum food safety, the U.S. Department of Agriculture recommends temperatures of 165° F for poultry, 145° F for fish, and 160° F for ground beef, lamb, and pork.