In a bowl, combine tomato juice, egg, bread crumbs, dried tomato, Worcestershire sauce, celery salt, hot sauce, and garlic if desired. Add ground beef and gently mix with a fork just until ingredients are evenly blended.
Divide meat mixture into 4 equal portions. With your hands, shape burgers into 3/4/-inch-thick patties.
If making ahead, lay patties side by side on a plate, cover airtight, and chill up to 24 hours.
Prepare a charcoal or gas grill for direct, medium-high heat (the charcoal or gas flame is directly under the cooking area, and you can hold your hand just above grill level only 3 to 4 seconds; click here for details).
For charcoal grill: Before you put the grill over the hot coals, brush it with a generous coat of oil. For gas grill: When hot, brush grill with a generous coat of oil.
In a small bowl, with a fork, mix mayonnaise and horseradish to blend.
Lay patties on well-oiled grill (close lid if using gas) and cook 5 minutes. Turn burgers with a wide spatula and continue cooking until meat is no longer pink in center (cut to test), about 5 minutes longer.
Spread hamburger buns with horseradish mayonnaise. Transfer burgers to buns and garnish to taste.