Cook shallot in oil with salt and pepper over moderate heat until golden, (2 to 3 minutes) then transfer to a bowl. Add turkey and minced red pepper and mix gently but thoroughly.
Divide turkey into 8 equal mounds. Pat 1 mound into a 4-inch patty and top with 1 ounce of cheese, then put a second mound on top, patting it onto the other patty to seal and shape into a single patty.
Grill burgers until no longer pink (about 10 minutes total).
To make the mayonnaise: Puree the roasted red peppers with water, vinegar and salt in a blender, then blend in the mayonnaise.
Cut the rolls in half horizontally (if necessary) and spread cut sides with roasted red pepper mayonnaise. Serve burgers on rolls with arugula.