1. To prepare burgers, pulse bread in a food processor to coarse crumbs, about 4 pulses. Transfer to a bowl. Place 1 1/2 pounds shrimp into food processor; add egg white, onion, Old Bay, and pepper, and pulse until there is a mix of finely minced pieces and coarsely chopped pieces, about 7 pulses. Add mixture to breadcrumbs.
2. Chop remaining 1/2 pound shrimp into 1/2-inch pieces; add to breadcrumb-shrimp mixture. Add 1/4 cup chopped parsley, and mix until combined. Shape mixture into 6 (1/3-cup) balls. To prepare sauce, combine mayonnaise, sour cream, lemon zest, lemon juice, capers, if desired, horseradish, and 1 teaspoon parsley. Reserve.
3. Heat 1 1/2 teaspoons oil in a medium nonstick skillet over medium-high heat until oil begins to shimmer, about 1 minute. Place 3 shrimp balls in pan, flattening to 1/2-inch thickness. Reduce heat to medium; cook until edges turn pink, and burger looks golden, about 3 to 4 minutes. Turn over and cook 3 minutes or until done. Repeat with remaining oil and shrimp balls.
4. Spread about 2 tablespoons sauce on each bun bottom. Top each with shrimp burger, lettuce leaf, and tomato slice. Cover with bun tops.