Divide the ground beef into eight equal portions and lightly form into balls. Flatten the meat into patties and press a divot into 4 of them ? these will be the bottom of the burgers. Place 1 tablespoon of the herb cheese into each divot. Cover with a second patty, pressing to seal around the cheese to the edges and taking care there are no air pockets. Repeat to make 4 burgers total.
Heat a griddle or grill over medium-high heat. Brush lightly with oil. Evenly sprinkle the patties all over with salt and pepper. Cook the burgers, turning once, until well-browned and an instant-read thermometer inserted sideways into the burger registers 145 degrees F, 8 to 10 minutes. Put the burgers on the bottom buns and top with lettuce and tomato. Cover with the top buns and serve.