Heat grill to medium-high. Once it is hot, clean the grill grate with a wire brush. Just before grilling, oil the grill grate.
Combine the avocado, basil, and ¼ teaspoon each salt and pepper in a small bowl.
Gently mix together the beef and ½ teaspoon each salt and pepper with your hands in a separate bowl until just combined. Form the beef 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, covered, until an instant-read thermometer inserted in the center of a patty registers 140° F, 3 to 4 minutes per side for medium-rare.
Top with the cheese and grill, covered, until the cheese melts, 30 seconds to 1 minute more.
Stack the burgers, tomato slices, and avocado mixture on the buns.