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.