Preheat the oven to 350ºF. Coat an 8 or 9 inch baking pan with butter.
In the bowl of a stand mixer fitted with the paddle attachment, beat the butter and brown sugar until fluffy. Add the eggs and beat well. Beat in the chocolate, beets, buttermilk, and vanilla. Add the flour and soda and beat until smooth. Pour the batter into the prepared pan and smooth the top. Bake 35 to 40 minutes or until a toothpick inserted in the center comes out clean. Let cool in the pan 5 minutes before removing it to a rack to cool completely.
To make the frosting, beat the cream cheese, confectioners' sugar, beets, and vanilla in the bowl of a stand mixer fitted with the paddle attachment until smooth. Cut the cooled cake in half horizontally and spread ⅓ of the frosting over the bottom layer. Replace top layer and spread the remaining frosting over the top and sides of the cake. Place the finished cake in the refrigerator for 30 minutes before slicing. The cake will keep, well covered, in the refrigerator for up to 4 days.