Run the lemon or lime wedge around the rim of a highball or pint glass to moisten. Mix the celery salt and coarse salt on a small plate and dip the moistened rim into the salt mixture to coat. Set the glass aside.

Make the Bloody Mary

Fill the prepared glass about three-quarters full with ice cubes (reserve a few cubes if you prefer less dilution).

In a cocktail shaker or large mixing glass (no need to shake vigorously like a shaken cocktail), add the vodka, tomato juice, prepared horseradish, Worcestershire sauce, lemon juice, lime juice, hot sauce, celery salt, freshly ground black pepper, and smoked paprika if using.

Stir the mixture gently for 15–20 seconds with a bar spoon or long spoon until well combined and slightly chilled. Taste and adjust seasoning: add more hot sauce for heat, more lemon/lime for brightness, or a pinch more celery salt for savory depth.

Pour the mix over the ice in the prepared glass. If you prefer a smoother texture, double-strain through a fine mesh into the glass to remove any horseradish bits.

Garnish and serve

Skewer the olives (and optional garnishes like a bacon strip or pickled bean) on a cocktail pick and rest it across the rim or drop it into the drink.

Place the celery stalk into the drink as a stirrer and add the lemon wedge on the rim. Serve immediately.

Optional: provide extra hot sauce, Worcestershire sauce, and celery salt at the table so each guest can adjust to taste.