Roughly chop the tomatoes and pass through a food mill into a 4-quart stainless steel mixing bowl. Stir in the balsamic vinegar and the garlic, and then whisk in the olive oil. Season with salt and pepper to taste, and set aside.
Add the crab, mayonnaise, chives and lemon juice to a large mixing bowl and mixing gently, combine well. Season with salt and pepper to taste, and blend in 1 tablespoon olive oil until just combined. Form the crab mixture into 4 round patties, approximately 1 1/2 inches high by 1 1/4 inches wide. Cover and refrigerate for 1 hour. Add 1 tablespoon of olive oil to a 10 inch non-stick saute pan, and place over medium-high heat. Meanwhile, coat the top and bottom of each crabcake with bread crumbs, transfer to the hot pan, and saute until golden brown. Turn and repeat until crabcake is browned on both sides (note: crabcakes are very delicate, so please be gentle when handling them). To serve, stack the basil leaves and slice thinly. Halve the avocado, remove the pit, and using a spoon, invert the half and insert the spoon between the flesh and the skin gently. Jimmy the spoon gently, working deeper and deeper until the flesh is free from the skin. Dice avocado into medium cubes. Ladle the soup into 4 chilled serving bowls and add one quarter of the avocado to each. Place 1 crabcake in the center of the bowl and sprinkle with chiffonade of basil. Garnish each crabcake with a basil top, and serve.
Recipe courtesy of Terrance Brennan, Picholine, New York, NY