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 1 hour ahead of serving. (A freezer gives a heavy frost.)
Place 1 slice of green tomato in each glass. Halve or quarter the bocconcini if they are larger than 1 inch in diameter. Thread a bocconccini, 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 take a leaf of basil per glass, and rub it around the glass rim, before placing on top of each glass, near skewer. Serve immediately.
*Cook's Note: It is impossible to give a specific amount for the tomatoes because how much "water" a tomato releases depends on many factors, including the growing season and the variety. Do not let the tomato pulp go to waste. Use this recipe as an excuse to start your quick tomato sauce season. You can freeze or can your sauce and then have it on hand for quick meals.
Bring 2 large pots of water to a boil. Lower the tomatoes into the water, blanch for 45 seconds to 1 minute. Place blanched tomatoes on a baking sheet, and peel the loosened skins. With a colander over a bowl, squeeze the seeds and innards out of each tomato. Set aside and refrigerate the clear strained liquid for other uses. Slice the peeled tomatoes into 1-inch chunks, and place chunks into blender. Puree thoroughly. Pour into bowl.