Put the minced shallot in a small strainer and rinse under cold water. (This will remove some of the raw onion flavor and help the shallot retain its crunch.) Pat dry, then transfer to a small bowl and add the celery, radish, and salt and pepper to taste.
Combine the vodka, tomato juice, the juice of one lemon half, the bitters, horseradish, Worcestershire sauce and hot sauce in a small pitcher and stir. Add salt and pepper to taste.
Add some crushed ice to 2 chilled glasses. Don't overload the Bloody Marys with ice so the flavors remain intense, not diluted. Add a hearty spoonful of the shallot mixture, then pour in the tomato juice mixture, filling the glasses almost to the top. Add another spoonful of the shallot mixture. Top each drink with a splash of vodka, like a "floater," and a squeeze of the remaining lemon half. Serve immediately.