Note: If buying shell-on shrimp, get about 1 1/2 pounds. Panko breadcrumbs are larger and lighter than traditional breadcrumbs, which could be substituted. From "Cook's Country Eats Local," by America's Test Kitchen.
For the tartar sauce: In a bowl, combine 3/4 cup mayonnaise, pickles plus brine, shallot, capers and 1/4 teaspoon pepper. Refrigerate until needed.
For the burgers: Pulse breadcrumbs in food processor until finely ground, about 15 pulses; transfer to shallow dish.
Place a third of shrimp (1 cup), 2 tablespoons mayonnaise, 1/4 teaspoon pepper, 1/8 teaspoon salt and cayenne in now-empty processor and pulse until shrimp are finely chopped, about 8 pulses. Add remaining 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 green onions.
Divide shrimp mixture into 4 (3/4-inch-thick) patties (about 1/2 cup each). Working with 1 patty at a time, dredge both sides of patties in breadcrumbs, pressing lightly to adhere; transfer to plate.
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.