Preheat the oven to 350 degrees F. Spray an 8-inch cake pan or Bunt pan with nonstick cooking spray.
Place the grated ginger in 2 layers of cheesecloth, or in 2 layers of paper towels. Squeeze to extract the juice, and set both the ginger and juice aside.
In a large bowl, whisk together the flour, pumpkin pie spice, baking powder, baking soda and juiced grated ginger.
In a separate bowl, whisk the egg with the brown sugar, applesauce, canola oil and vanilla until well combined. Stir the wet ingredients into the dry ingredients until thoroughly incorporated. Scrape the batter into the prepared cake pan, leveling the top with a spatula. Bake until a toothpick inserted in the center comes out mostly clean with a few crumbs clinging to it, 40 to 45 minutes. Remove from the oven. When cool enough to handle, invert the cake onto a wire cooling rack.
For the glaze, stir together the powdered sugar with the reserved ginger juice. If the glaze is too thick, just add a little water to form a pourable but still thick consistency.
When the cake is completely cooled, about 1 hour, pour the glaze onto the center and use an offset spatula to help the glaze run down the sides. Sprinkle the edges with the chopped walnuts to garnish.