Brush the bread with olive oil, then bake in a hot oven or under the broiler until crisp and golden brown, a few minutes on each side, then set aside.
Cook about half of the tomato puree with the salt for about 5 minutes on low heat. Then, add the remaining puree, balsamic vinegar, pepper to taste, and the vodka.
Blend the mozzarella with about 2/3 of the basil, a pinch of white pepper to taste, until it forms a smooth cream; place in the bottom of four wide-bottomed martini glasses.
Add a layer of the tomato mixture, and garnish with a stick of celery, a toasted bread crouton, and a few leaves of the reserved basil.