Pour the celery salt and coarse black pepper onto a small plate and spread into a shallow ring. Run a lemon wedge around the rim of each highball or pint glass to moisten. Dip the rim into the salt and pepper mixture, twisting to coat evenly. Set glasses aside.

Make the Bloody Mary mix

In a large pitcher, combine 4 cups tomato juice, 2 tablespoons prepared horseradish, 2 teaspoons Worcestershire sauce, 2 tablespoons fresh lemon juice, 1 teaspoon hot sauce, 1 teaspoon ground black pepper, 1 teaspoon celery salt, 1 teaspoon smoked paprika, and 1 teaspoon salt.

Whisk vigorously until ingredients are fully incorporated and the mixture is smooth. Taste and adjust seasoning: add more hot sauce for heat, more horseradish for bite, or additional lemon for brightness.

Refrigerate the mix for at least 15 minutes to let flavors meld. The mix can be made up to 24 hours ahead; stir before using.

Assemble cocktails

Fill each prepared glass about two-thirds full with ice (about 1 cup per glass).

For each serving, pour 2 ounces (about 1/4 cup) vodka over the ice.

Top each glass with about 1 cup (8 ounces) of the Bloody Mary mix, leaving a little room at the top.

Stir gently with a long spoon to combine vodka and mix. Taste and adjust by adding a few drops of hot sauce, a squeeze of lemon, or a pinch more celery salt if needed.

Garnish and serve

Skewer two olives (and an optional cherry tomato or small pickle) on a cocktail pick. Place a lemon wedge on the rim and add a celery stalk to each glass as a stirrer.

Serve immediately with a straw if desired.