To make the burgers: Put oven rack in middle position and preheat oven to 350 degrees F.
Cook shallot in oil with salt and pepper in an 8-inch skillet over moderate heat, stirring occasionally, until golden, 2 to 3 minutes, then transfer to a bowl. Add turkey and mix gently but thoroughly.
Turn out turkey mixture onto a sheet of waxed paper and divide into 8 equal mounds. Pat 1 mound into a 4-inch patty and top with 1 piece of cheese, then put a second mound on top, patting it onto the other patty to seal and shape into a single patty.
Heat an oiled well-seasoned ridged grill pan over moderately high heat until hot but not smoking, then grill burgers, turning over once, until just cooked through (no longer pink), 8 to 9 minutes total.
While burgers are cooking, heat rolls on a baking sheet in the oven until crusty, about 5 minutes. Transfer rolls to a rack to cool slightly.
To make the mayonnaise and assemble the burgers: Puree the tomatoes with water, vinegar and salt in a blender or mini food processor, scraping down the sides as necessary, then blend in the mayonnaise.
Cut the rolls in half horizontally (if necessary) and spread cut sides with tomato mayonnaise. Serve burgers on rolls with lettuce.