Cut salmon into 1/4-inch dice, then stir together with spinach, scallions, ginger, salt, and pepper in a large bowl until well combined.
Beat together egg white and soy sauce in a small bowl and stir into salmon mixture, then form into 4 (1/2-inch-thick) patties.
Heat a 12-inch nonstick skillet over moderate heat until hot and lightly brush with oil. Cook patties, carefully turning once, until golden brown and cooked through, 6 to 7 minutes total.
Serve each burger topped with 1 1/2 teaspoons pickled ginger.
