Preheat grill to medium-high. Cut cheese into 4 equal chunks and set aside.
In a large bowl combine the ground beef, pesto and a sprinkle of salt and pepper. Be sure not to over-mix.
Shape the beef into 8 thin patties. Take a piece of cheese and sandwich it between two thin patties and then seal the edges to make one patty, making sure the cheese is completely hidden inside the patty. Repeat with the remaining patties and cheese, until you have four cheese-filled patties.
Grill burgers on pre-heated grill until cooked through. Serve on buns and enjoy!