Step 1

Place 3 tablespoons butter in a small microwavable bowl. Microwave on HIGH until melted, about 25 seconds. Combine melted butter, half of shrimp, and egg in bowl of a food processor. Pulse until finely chopped and mixture starts to clump together, about 12 times. Gently stir together shrimp mixture, panko, scallions, 1 1/4 teaspoons salt, 1/2 teaspoon Old Bay, lemon zest, 1 tablespoon lemon juice, and remaining shrimp in a large bowl. Shape mixture into 4 (3 1/2-inch) patties (about 5 1/2 ounces each). Transfer patties to a parchment paper–lined baking sheet; cover and chill until firm, about 15 minutes.

Step 2

Meanwhile, stir together mayonnaise, mustard, hot sauce, remaining 1/2 tablespoon lemon juice, remaining 1/4 teaspoon salt, and remaining 1/4 teaspoon Old Bay in a small bowl; set aside.

Step 3

Melt remaining 2 tablespoons butter in a large nonstick skillet over medium-low. Add shrimp patties, and cook until golden brown and cooked through, 5 to 6 minutes per side. Spread mayonnaise mixture evenly on cut sides of buns. Top bottom bun halves evenly with shrimp patties, lettuce, tomato slices, and avocado slices; cover with top bun halves.