Using your hands, combine beef, 3 teaspoons salt, and the pepper in a bowl. Shape into 6 balls. Scoop a heaping tablespoon from center of each; reserve.
Fill each burger with 1/2 ounce mozzarella, 1/2 teaspoon herbs, and 4 tomato halves; season with salt.
Press the reserved meat over filling, pinching to seal. Gently flatten into patties (about 5 inches in diameter). Refrigerate, covered, until cold, about 1 hour.
Preheat grill to medium-high (if using a charcoal grill, coals are ready when you can hold your hand 5 inches above grill for just 3 to 4 seconds). Brush grill and burgers with oil. Grill, flipping once, about 4 minutes per side for rare or 10 minutes per side for well-done. Toast buns, if desired. Serve burgers on buns.