Preheat oven to 350°F. Butter and flour a round 8-inch cake pan (at least 2-inch deep). Tap out an excess flour. Line bottom with parchment paper. Set aside.
In a large bowl, whisk together all-purpose flour, cake flour, baking powder, baking soda, and salt. Set aside.
Place eggs in a medium-large bowl. Using an electric hand mixer (or stand mixer fitted with a whisk attachment), beat eggs until foamy. Add cream, brown sugar, grated ginger, and vanilla. Beat for 8-10 minutes until thickened. Mixture should almost resemble soft peaks.
Pour whipped egg mixture into dry flour mixture. Fold together to combine until there are no longer any dry streaks of flour. Fold in chopped candied ginger.
Transfer batter to prepared cake pan. Spread into an even layer. Bake for 50-60 minutes, rotating pan midway through baking, until toothpick inserted in center of cake comes out clean. Allow to cake to cool in pan for 15 minutes.
Run a mini offset spatula (or knife) along the edges of the cake to help loosen from pan. Unmold cake and allow to cool to room temperature on wire rack.