You will need enough tomato water to fill your martini glasses, so measure their capacity, probably something between 5 and 9 ounces.

Chill the martini glasses in the freezer, if there is room, or refrigerate an hour ahead of serving. (A freezer gives a heavy frost.) Peel, seed, and chop the 3 to 4 pounds tomatoes. Season lightly with salt and pepper, place in a sieve suspended over a bowl and refrigerate for several hours. As the tomatoes release their juice, it will fall into the bowl, giving you an almost gin-clear liquid with lots of flavor. Season the tomato water to taste with salt and pepper and refrigerate until very cold. Reserve the tomato pulp for another use.

Halve or quarter the bocconcini if they are larger than 1 inch in diameter. Thread a tomato wedge, a bocconcini, and a cherry tomato onto each of 4 wooden skewers 4 to 5 inches long. Season with salt and pepper and drizzle with basil oil, if desired. Working quickly, divide the tomato liquid among the chilled martini glasses. Balance the skewers on top of the glasses, then sprinkle the basil chiffonade over the skewers. Serve immediately.