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 an outdoor grill for medium-high heat, and lightly oil the grate.
Cook on the preheated grill until the patties are no longer pink in the center, about 5 minutes per side for well done. An instant-read thermometer inserted into the center should read 160 degrees F (70 degrees C). Place a slice of gouda cheese on each patty; continue grilling until cheese has melted, about 1 minute. While the turkey patties are cooking, place the pineapple rings on the grill until softened, 2 to 3 minutes per side.
Assemble the burgers by placing a turkey patty on half of an onion roll, then top with the grilled pineapple. Divide the spinach leaves evenly among the burgers, then top with the other half of the onion roll.