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.
½ cup finely chopped crystallized ginger, 1/2 cup plus 1 tablespoon all-purpose flour
In a large bowl, stir together the melted butter, applesauce, sugar, and eggs until light and creamy. Add the molasses and buttermilk.
¼ cup unsalted butter, ¼ cup unsweetened applesauce, ½ cup sugar, 1 large egg, 1/3 cup molasses, ½ cup 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.
1/2 cup plus 1 tablespoon all-purpose flour, ¼ teaspoon baking soda, 1 1/2 teaspoon ground ginger, ½ cup white whole wheat flour, 1 teaspoon ground cinnamon
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.