Whisk together the bread crumbs, egg white, onion, soy sauce, minced ginger, five-spice powder, ground ginger and pepper in a large bowl. Mix in turkey until just combined. Shape turkey mixture into four patties.
Preheat the barbecue for medium-high heat and lightly oil the grate.
Cook until the patties are no longer pink in the centre, about 5 minutes per side for well done. Place a slice of Gouda cheese on each patty; continue cooking until cheese has melted, about 1 minute. While the turkey patties are cooking, place the pineapple rings on the barbecue until softened, 2 to 3 minutes per side.
Assemble the burgers by placing a turkey patty on half of an hamburger roll and topping with the pineapple. Divide the spinach leaves evenly among the burgers then top with the other half of the roll.