Mix together the mayonnaise, Dijon and vinegar. Place tuna in a large bowl. Fold in the mayonnaise mixture, olives, basil, capers and shallots and season with salt and pepper to taste. Chill the mixture well. Divide the mixture into fourths and mold into burgers. Chill again for 30 minutes. Grill for 2 to 3 minutes on each side.
Place tomatoes, lemon juice and garlic in the blender and blend until smooth. Add the mayonnaise and blend until just combined. Season with salt and pepper to taste.