Step 1

Mix hard-boiled eggs, onion, mayonnaise, ketchup, and ½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt in a small bowl until combined. Set egg salad aside.

Step 2

Cut one third of shrimp into ½” pieces; finely chop remaining shrimp (or pulse in a food processor). Transfer all shrimp to a medium bowl. Mix in potato starch, pepper, 1 Tbsp. beaten eggs, and ½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt. Using oiled hands, divide mixture into 4 portions, then shape into ½”-thick patties. Place on an oiled baking sheet and chill 1 hour.

Step 3

Place panko, flour, and remaining beaten eggs in 3 separate shallow bowls. Working with 1 patty at a time, dredge in flour, shaking off excess. Dip in eggs, letting excess drip back into bowl, then coat in panko, pressing to adhere before shaking off excess.

Step 4

Pour oil into a large heavy skillet, preferably cast iron, to come 1” up sides. Clip thermometer to sides of pan. Heat oil over medium until thermometer registers 350°. Working in 2 batches, carefully slide patties into oil and fry, turning halfway through, until golden brown and cooked through, about 3 minutes per side. Transfer to a wire rack set inside a rimmed baking sheet; season with salt.

Step 5

Melt 1 Tbsp. butter in another large skillet over medium heat. Working in 2 batches and adding remaining 1 Tbsp. butter between batches, toast buns, cut sides down, until golden brown, about 2 minutes. Transfer to plates.

Step 6

Spread each bottom bun with 1 Tbsp. wasabi mayonnaise. Top each with some cabbage and a patty. Spoon reserved egg salad over and close burgers.Do ahead: Egg salad can be made 1 day ahead; cover and chill. Patties can be formed 12 hours ahead; cover and keep chilled.