Make the tomato vodka: Cut each of the tomatoes into 8 pieces and put them in a large glass jar. Add the vodka, stir to combine, and cover. Put in a cool, dark place for at least 5 days and up to 7 days, stirring every day. After 5 to 7 days, strain through a fine-mesh strainer and discard the solids. To store, pour the tomato vodka back into its original bottle and keep in a cool, dark place.
Make the Bloody Marys: Put the cherry tomatoes, hot sauce, Worcestershire sauce, lemon juice, and salt in a blender, cover and blend on high speed for 1 1/2 minutes. Fill an ice cube tray with about 2 cups of the juice and freeze overnight. Pour the remaining juice into a lidded container and reserve on the counter overnight. (Refrigeration will destroy these particular flavors and aromas.)
Just before serving, put 3 frozen cubes in each of 4 Collins glasses and add 1 1/2 ounces tomato vodka to each glass. Stir the reserved juice, pour 4 ounces into each glass, and serve immediately.