Cutting the fish by hand keeps the texture of the patties light and fluffy.
Cut fish into 1/4-inch pieces, then coarsely chop until it holds together. Mix with eggs, 1/3 cup mayonnaise, breadcrumbs, capers, chives, 1 teaspoon salt, and cayenne. Form into 8 patties, each about 3 1/2 inches in diameter. Refrigerate, covered, at least 1 hour and up to 8 hours.
Heat 1 tablespoon oil in a large, heavy skillet over medium-high. Cook 4 burgers, flipping once, until golden brown and cooked through, about 10 minutes. Repeat with remaining oil and burgers.
Combine remaining 1/2 cup mayonnaise and mustard. Brush split sides of buns with oil; lightly toast in skillet. Sandwich burgers in buns with mayonnaise mixture, red onions, and watercress.