Preheat oven to 350 degrees. Grease an 8x8-inch baking pan, and line with parchment paper. Set aside.

In a small bowl, stir together the candied or crystallized ginger with 1 tablespoon of the all-purpose flour. This will prevent the ginger pieces from sticking together and help keep them from all sinking to the bottom of the cake.

In a large bowl, stir together the melted butter, applesauce, sugar, and eggs until light and creamy. Add the molasses and buttermilk.

In a separate bowl, whisk together the flours, baking soda, ginger, and cinnamon. Add the dry ingredients to the wet, and stir until just combined. Fold in the ginger pieces.

Pour batter into the prepared pan. Baked for 25-35 minutes, until the center is set and a toothpick inserted into the middle comes out with only a few crumbs clinging to it.

Remove from oven and allow to cool in the pan. Using the parchment as a sling, remove the cake from the pan and cut into pieces. Serve dusted with powdered sugar or with lightly-sweetened whipped cream.