For the cake: Preheat the oven to 375 degrees F. Lightly grease a 10-cup bundt pan with nonstick cooking spray. Dust with cocoa powder and tap out the excess.
In a medium bowl, combine the all-purpose flour, whole wheat flour, baking soda, baking powder and salt.
Melt the oil and chocolate together over a double boiler, whisking to combine. Remove from the heat, and add the eggs one at a time, beating after each addition. Stir in the pureed beets. Add the dry ingredient mixture in three parts, stirring to just combine after each addition. Pour the batter into the prepared pan, smooth the top and place in the oven. Bake until a toothpick inserted in the center of the cake comes out clean, 45 to 55 minutes.
For the glaze: In a medium bowl, whisk together the confectioners' sugar, milk and vanilla extract until smooth. Drizzle over the cooled cake.