Leave the pointed tips on the green beans, but snap off the stem tips. Wash. Bring a large pot of water to a boil. Add the sugar and 1 tablespoon salt. Drop in the green beans and cook until crisp-tender, about 6 to 7 minutes. Drain the beans and plunge them into ice water to stop cooking. Drain well and refrigerate.
When ready to serve, toss the green beans with 1/3 cup of the dressing in a large mixing bowl. Taste and add more dressing or salt as needed. Pile the beans high in the center of a large white platter. Arrange the tomato slices overlapping around the edge. Sprinkle the tomatoes with remaining salt and drizzle 3 tablespoons of the dressing on top of them. Garnish with basil sprigs. Serve immediately.
Turn on the processor fitted with the steel blade and drop the garlic and shallot down the feed tube onto the spinning blade to mince. Add the water, salt, pepper, mustard, and sugar. With the processor running, slowly drizzle in the oil. Add the basil leaves and coarsely chop with several on/off pulses.