Preheat grill to medium-high. Combine cucumber, celery, pepperoncini, and sugar in a medium bowl; season with salt and pepper and toss to combine. Form beef into 4 patties, then season them generously with salt and pepper. Spread inside of buns with butter.
Add patties to grill. For medium-rare burgers, grill patties, uncovered, for 4 minutes; add cheese and grill for 4 to 5 minutes more. Transfer burgers to a plate. Add buns, buttered-side down, to grill until charred, about 1 minute. Sandwich the burgers in the buns and top with celery slaw. Serve with chips or salad, if desired.
Before her diagnosis, Caroline wrote a book on cakes called Cake Magic!. She started developing a birthday cake using her gluten-free mix found in that book. Check out other recipes she’s developing for her new life—and the stories behind them—on her blog, The Wright Recipes. Her next book, Catalan Food, written with chef Daniel Olivella, comes out in early September from Clarkson Potter.