Crispy Golden Shrimp Burgers

Name: Crispy Golden Shrimp Burgers
Ingredients: 4 hard-boiled eggs, coarsely chopped; ¼ medium onion, finely chopped; 4 Tbsp. mayonnaise (preferably Kewpie); 2 Tbsp. ketchup; 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt, divided, plus more; 12 oz. shrimp, peeled, deveined; 2 tsp. potato starch or cornstarch; ½ tsp. freshly ground black pepper; 2 large eggs, beaten to blend, divided; Vegetable oil (for frying; about 4 cups), plus more; 1 cup panko; ⅓ cup all-purpose flour; 2 Tbsp. unsalted butter; 4 brioche buns, split; 4 Tbsp. wasabi mayonnaise (such as Kikkoman Wasabi Sauce); ¼ medium head of savoy cabbage, cored, thinly sliced; A deep-fry thermometer