Put the tomato slices in a large glass container or pitcher. Add the vodka (make sure it covers the tomatoes) and a pinch of salt; let infuse 6 to 12 hours at room temperature. Strain the vodka, then chill until ready to serve.
Meanwhile, prick the grape tomatoes with a skewer; toss with the balsamic vinegar in a bowl and marinate 4 hours at room temperature.
Drain the grape tomatoes, reserving the vinegar. Bring the vinegar to a boil in a nonreactive saucepan over medium heat. Reduce the heat and simmer until syrupy, about 20 minutes; let cool.
For each drink, drizzle some of the balsamic syrup into the bottom of a chilled martini glass. Muddle 2 to 3 basil leaves with a splash of tomato-infused vodka in a cocktail shaker. Add another 1 1/2 ounces tomato-infused vodka and 1 cup ice; shake and strain into the glass. Skewer 2 marinated grape tomatoes, 2 basil leaves and a mozzarella ball on a cocktail pick and set in the martini.