How to Make It
Place water, vinegar, sugar, and 2 tablespoons of the salt in a small pot. Heat over high, undisturbed, until sugar and salt dissolve, about 2 minutes. Place cucumbers in a small heatproof bowl. Pour mixture over cucumber; let stand at room temperature 20 minutes (or covered, in refrigerator up to 24 hours).
Place salmon, egg white, hoisin, gochujang, and remaining 1 teaspoon salt in a food processor; pulse until finely chopped, about 30 pulses. Shape mixture evenly into 4 (1-inch-thick) patties. Place patties on a plate; refrigerate 15 minutes.
Heat oil in a large nonstick skillet over medium. Add patties; cook until lightly browned and a thermometer inserted in thickest portion of patties registers 145°F, 3 to 4 minutes per side.
Spread mayonnaise evenly over cut sides of toasted buns. Layer bottom bun halves evenly with cabbage, patties, pickled cucumber, onion, and radishes. Replace top bun halves.