Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 9x13 inch pan.
In a large bowl, mix 1 3/4 cup flour, 1 1/2 sugar, cocoa, baking powder and soda.
Whisk in buttermilk, 1/2 cup butter, eggs, and vanilla. Stir in shredded beets and pour batter into cake pan.
Bake at 350 degrees F (175 degrees C) for 45 minutes or until a toothpick inserted in the center of cake comes out clean.
For the frosting: Combine milk and 5 tablespoons flour in a small saucepan, heat to a boil. Stir constantly for 2 to 3 minutes. Stir in 1 cup sugar, 1/3 cup butter and shortening. Cool resulting mixture to room temperature and beat for 10 to 12 minutes until light and fluffy. Spread on cooled cake. This frosting is also excellent on spice or applesauce cake.