Rödbeteskaka i röd sammet

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1. 1
   Peel and dice the beet. Transfer to a blender and purée until completely smooth. Reserve in the refrigerator until ready to bake. Beet purée may be made up to 1 day in advance.
2. 2
   Heat the oven to 350°F. Grease two 9-inch round cake pans with nonstick cooking spray and line with parchment paper.
3. 3
   In the bowl of a stand mixer fitted with the paddle attachment, cream the butter and sugars together until fluffy, 2 to 3 minutes. Add the eggs one by one, beating to combine completely between additions, then scrape down the sides of the bowl.
4. 4
   In a small bowl, stir together the cocoa powder with 4 tablespoons beet purée to make a paste. Add to the stand mixer and beat to combine.
5. 5
   In a measuring cup, stir together the salt, vanilla bean paste, and buttermilk. In a medium bowl, sift together the cake flour and baking powder. Add ½ cup flour mixture to the batter and mix on low until just combined, then add ¼ cup buttermilk mixture, mixing until just combined. Repeat until both mixtures have been used up. Scrape down the bowl with a rubber spatula, then gently fold in the vinegar and stop stirring.
6. 6
   Divide the batter evenly between the prepared cake pans. Bake, rotating halfway through, for 30 minutes, or until a toothpick inserted into the center comes out clean. Let cool in pans for 10 minutes, then invert onto a wire rack, peel off the parchment paper, and cool completely.
7. 7
   Combine flour and milk in a small saucepan over low heat. Stir constantly until the mixture thickens, then remove from the heat and let cool complete.
8. 8
   Beat the coconut sugar, butter, and vanilla bean paste in the bowl of a stand mixer fitted with the paddle attachment on medium speed until light and fluffy, 2 to 3 minutes. Add the cream cheese and beat again until fluffy, 1 to 2 minutes more. Add the cooled flour mixture and beat again until the frosting is silky and spreadable, a final 1 to 2 minutes.
9. 9
   When the cakes are completely cool, set one on a serving plate. Frost the top of it, then place the second cake on top, bottom side up. (This will give you a flat top surface.) Frost the top and sides with the remaining icing, then serve.