Make shrimp patties. Set aside 4 ounces of the shrimp and roughly chop by hand into small chunks. With the remaining 12 ounces, pulse shrimp in a food processor with garlic powder, salt, pepper, paprika, chili powder, and dry mustard until a paste forms. Combine shrimp paste with chopped shrimp, egg white, and all-purpose flour, mixing until well combined. Shape shrimp mixture into 4 patties.

Place panko in a shallow bowl. Coat patties evenly with panko, making sure to cover all sides. Place burgers on a baking sheet and chill in the fridge.

Meanwhile, make tartar sauce. Combine chopped eggs, mayonnaise, chopped onion, cornichons, dill, lemon juice, hot sauce, salt, and pepper in a large bowl until well combined. Set aside.

Make honey mustard aioli. Combine mayonnaise with mustard, honey, and lemon. Season with salt and pepper and set aside.

Season shredded cabbage with olive oil, lemon, and salt, tossing to combine. Set aside.

Heat enough oil in a large saute pan to cover the bottom of the pan over medium heat. Cook the shrimp patties until golden brown on both sides, about 5-6 minutes. Remove from heat.

Assemble burgers. Smear bottom brioche bun with honey mustard aioli and top with cabbage, shrimp patty, tartar sauce, and remaining bun. Serve immediately.