1. Beat butter, tarragon, thyme, and parsley with spoon in bowl until combined; transfer to sheet of plastic wrap. Using wrap, shape butter into a 3x2-inch log. Cover and refrigerate until firm, 3 hours.
2. Cut chilled herb butter into 8 equal slices. (Wrap 3 slices and refrigerate for another use.) Cut each slice into small pieces.
3. Divide beef into quarters. Gently shape each quarter into a patty. Split one patty in half horizontally. Sprinkle pieces from 1 slice herb butter over split side of patty. Replace top half of patty, gently pinching edge to seal. Gently shape until 1 inch thick. Transfer to large plate. Repeat with remaining beef and herb butter. Chill patties in freezer 15 minutes.
4. Heat grill. Sprinkle salt and pepper over both sides of patties. Grill burgers 7 to 8 minutes per side, or until meat thermometer reaches 160 degree F when inserted 1 inch from edge of each burger for medium.
5. While burgers are grilling, spread remaining slice of herb butter over cut sides of rolls. Grill rolls cut side down, 1 to 2 minutes, until toasted.
6. Place lettuce leaves on bottom halves of rolls. Top with burgers, then tomato slices. Serve with French fries or chips, if desired. Makes 4 burgers.