Pulse shrimp in food processor until there is an even mix of finely minced and coarsely chopped pieces of shrimp, about 7 pulses, scraping down sides of bowl as needed.
Whisk panko, mayonnaise, scallion, parsley, lemon zest, salt, pepper, and cayenne together in large bowl. Gently fold in processed shrimp until just combined.
Scrape shrimp mixture onto small baking sheet. Divide mixture into 2 equal portions, gently flatten each portion into 1-inch-thick patty, and press shallow divot in center of each patty. Cover with plastic wrap and refrigerate for 15 minutes.
Heat oil in 10-inch nonstick skillet over medium heat until shimmering. Gently place burgers in skillet and cook until crisp and browned on both sides, about 4 minutes per sides. Serve burgers on buns.