In a large pitcher, pour 4 cups tomato juice. Add 1 tablespoon prepared horseradish, 2 tablespoons Worcestershire sauce, 2 tablespoons fresh lemon juice, 1 tablespoon fresh lime juice, and 1 tablespoon hot sauce.

Season and balance

Stir in 1 teaspoon ground black pepper, 1 teaspoon celery salt, 1 teaspoon smoked paprika, 1 teaspoon sugar (optional), and 1/2 teaspoon salt. Add the extra 1 teaspoon prepared horseradish only if you want a spicier mix. Taste and adjust: add more hot sauce, horseradish, lemon, or sugar to reach desired heat and brightness.

Chill

Cover the pitcher and refrigerate the mix for at least 30 minutes to let flavors marry. The mix can be made up to 24 hours ahead; stir or shake before using.

Cocktail and garnish

Prepare glass rim

Pour 1 tablespoon celery salt and 1 teaspoon coarse salt onto a small plate and mix. Rub a lemon wedge around the rim of each highball glass, then dip the rim into the celery/cane salt mixture to coat.

Assemble cocktail

Fill each rimmed glass with a generous handful of ice (about 1/2 cup ice per glass). Pour 4 ounces (1/2 cup) of chilled Bloody Mary mix into each glass, then add 2 ounces vodka per cocktail (for this recipe, total vodka is 8 oz to make four 2-oz servings). Stir gently to combine.

Garnish

Garnish each cocktail with a celery stalk, a lemon wedge, a pickle spear, and two olives on a cocktail pick. Add a slice of pickled jalapeño or pepperoncini if you like extra tang/heat.

Serve

Serve immediately with a small straw or stirrer. Offer additional hot sauce, Worcestershire, lemon, and celery salt at the table for guests to adjust to taste.