If you are using a juicer, juice the celery, beets, and bell pepper. Place the liquid into a pitcher or large mixing bowl and add the tomato puree.
If you are using a blender, put the celery, beets, bell pepper, and tomato puree into a blender. Pulse until you have a vegetable smoothie. Then make a sturdy pouch out of some folded cheesecloth and, over a pitcher or a large bowl, carefully pour the blended veggies into it. Gather the cheesecloth at the top to close and then squeeze. Really squeeze, many times, until all the juice runs out of the cheesecloth and all you have left inside is dry veggie pulp. If you don’t have cheesecloth, you can also pour the veggie smoothie into a fine-mesh sieve and use a wooden spoon or the bottom of a ladle to press down on the puree, squeezing the juice through until you have strained every bit into the pitcher or bowl. Set all your vegetable scraps aside.
Now that you have your vegetable juice, add 1 teaspoon of the horseradish, 1 teaspoon of black pepper, the smoked paprika, cayenne, juice of 1 lemon, and vinegar to the pitcher or bowl. Whisk together and taste, adjusting the spices (oh hello, horseradish and pepper) according to your cocktail preferences. I like mine spicy!
Place the Bloody Mary mixture in the refrigerator to chill, 15 minutes minimum or 2 hours maximum. When you’re ready to serve, fill glasses with ice. Pour in Bloody Mary mixture and add a celery stalk for garnish. Cut the remaining lemon into wedges and offer them to guests as well for extra citrus punch.