1. FOR THE TARTAR SAUCE: Combine all ingredients in bowl and refrigerate until needed.

2. FOR THE BURGERS: Pulse panko in food processor until finely ground, about 15 pulses; transfer to shallow dish. Place one-third of shrimp (1 cup), mayonnaise, pepper, salt, and cayenne in now-empty processor and pulse until shrimp are finely chopped, about 8 pulses. Add remaining two-thirds of shrimp (2 cups) to shrimp mixture in processor and pulse until coarsely chopped, about 4 pulses, scraping down sides of bowl as needed. Transfer shrimp mixture to bowl and stir in scallions.

3. Divide shrimp mixture into four 3/4-inch-thick patties (about 1/2 cup each). Working with 1 patty at a time, dredge both sides of patties in panko, pressing lightly to adhere, and transfer to plate.

4. Heat oil in 12-inch nonstick skillet over medium heat until shimmering. Place patties in skillet and cook until golden brown on first side, 3 to 5 minutes. Carefully flip and continue to cook until shrimp registers 140 to 145 degrees and second side is golden brown, 3 to 5 minutes longer. Transfer burgers to paper towel–lined plate and let drain, about 30 seconds per side. Spread tartar sauce on bun bottoms, then place burgers and lettuce on top. Cover with bun tops. Serve.