Preheat a grill or grill pan over medium-high heat. Combine the ground beef with 1/3 of the pesto and salt and pepper to taste in a large bowl until well incorporated.
Divide the meat mixture into four patties and grill to your desired doneness, anywhere between 3-4 minutes per side, adding 1 slice of cheese to each burger during the last 2-3 minutes of grilling.
Toast the buns on the grill then brush them lightly with the remaining pesto. Top the bottom half of each bun with a cheese-topped burger patty, a red onion slice and some basil leaves. Top with the remaning bun halves.