Preheat oven to 350°F. Coat a bundt pan with cooking spray.
In a medium bowl, whisk together all-purpose flour, whole-wheat flour, granulated sugar, cocoa, baking powder, baking soda, cinnamon, ginger, cardamom, nutmeg, allspice, and salt.
In a large bowl or the bowl of a stand mixer, blend 1 cup milk, pumpkin, and brown sugar in a large bowl with an electric mixer on low speed. Beat in eggs. Stir in applesauce and vanilla. Gradually add the dry ingredients, stirring until just combined. Pour the batter into the prepared pan.
Bake the cake for 60-75 minutes, until a toothpick inserted in the cake comes out with only a few moist crumbs attached. Let cool on a wire rack for 15 minutes. Remove from the pan and let cool completely on the rack, about 2 hours.
To glaze cake: Combine powdered sugar and 1 tablespoon milk in a small bowl, stirring until completely smooth. Place the cake on a serving plate and drizzle the glaze over the top. Let harden before serving.