Trim, core and quarter the fennel. Roughly chop 1/4 cup fronds. Thinly slice the fennel (use a mandoline if you have one) and combine with the red onion, fennel fronds, lemon juice and 1 tablespoon olive oil in a medium bowl. Toss and season with salt.
Combine the mayonnaise and cherry peppers in a mini food processor and puree until smooth. (Alternatively, finely mince the cherry peppers and stir into the mayonnaise.) Refrigerate until ready to serve.
Form the lamb into 4 thin patties, about 4 inches wide and 1/2 inch thick. Season on both sides with salt and pepper. Heat a large skillet over high heat until very hot and add the remaining 1 tablespoon olive oil. Add the burgers and cook until well browned, about 2 minutes per side for medium.
Add the arugula to the fennel salad and toss to combine. Serve the burgers on the buns; top with the mayonnaise mixture and fennel salad. Serve with potato chips.