Tomato Water
In a blender, purée the tomatoes, starting on low speed and increasing to high speed. Transfer the tomatoes to the fine-mesh strainer, and press down with a small ladle or spoon to extract all the juice from the pulp. Pour the juice into a 1-pint freezer-safe container and freeze until solid, about 4 hours.
Line a colander with a few layers of cheesecloth, and set it into a large bowl. Place the frozen block of tomato juice in the cheesecloth. Cover the colander with plastic wrap, place in the refrigerator, and allow the tomato block to drip as it thaws, up to 24 hours depending on the size of the frozen block. Pour the resulting clear tomato water into a glass jar. You should have a bit less than 1 cup (240 ml). (Store in the refrigerator for up to 5 days.)
Basil Syrup
In a small saucepan, bring the sugar and water to a boil over medium-high heat, stirring to dissolve the sugar. Remove from the heat and let cool to room temperature. Pour the syrup into a 1-cup (240 ml) jar and add the basil leaves. Let stand for about 24 hours, shaking occasionally. Strain the syrup. (Syrup can be stored in the cleaned jar and refrigerated for up to 2 weeks.)
Saline Solution
In a small saucepan, bring the water to a boil. Add the salt and stir until dissolved. Let cool completely, then transfer to a 1/4-cup (60 ml) glass bottle with a dropper lid or a small container. Close the lid and keep behind the bar or in the refrigerator for several months.
For Serving
Thread the mozzarella, tomatoes, and basil leaves on a skewer. Fill a coupe with ice cubes to chill.
In a shaker, combine 1/4 cup (60 ml) of the tomato water, the rum,
1 tablespoon of the syrup, the balsamic vinegar, the whey, and 8 drops of the saline solution. Fill the shaker with ice and shake the cocktail until well chilled.
Pour out the ice from the coupe and Strain the drink into the chilled glass. Add the garnish.
