Combine egg, bread crumbs, lemon juice, lemon zest, sesame oil, seafood seasoning, and black pepper in a bowl. Add shrimp and butter. Form mixture into small slider sized patties and refrigerate at least 30 minutes.
Heat oil in a skillet over medium heat. Cook patties until browned, 4 to 5 minutes. Flip and cook until second side is browned, 4 to 5 minutes more.