How to Make It
Preheat the oven to 350º. Wrap the beets in foil and bake for about 1 hour, until tender. Let cool slightly.
Peel the roasted beets and cut them into chunks. Transfer to a food processor and let cool completely. Puree until smooth. Measure out 1 1/3 cups of the puree; reserve any remaining beet puree for another use.
In a medium saucepan, bring the milk just to a boil with the butter. Remove from the heat. Whisk in the 1 1/3 cups of beet puree. In a medium bowl, sift the flour with the cocoa powder and salt.
Line a large rimmed baking sheet with parchment paper, allowing 1 inch of overhang on the two long sides. Add the eggs and the granulated and brown sugars to the bowl of a stand mixer. Set the bowl 2 inches above a saucepan of simmering water and beat until smooth and the sugars dissolve and the mixture is slightly warmed, about 4 minutes.
Transfer the bowl to the stand mixer fitted with the whisk. Beat the warm egg mixture at high speed until thickened and cooled, about 5 minutes. Scrape into a large bowl and, using a rubber spatula, gently fold in the dry ingredients just until no streaks remain. Fold one-third of the batter into the beet milk, then gently fold the beet mixture into the batter in three additions.
Scrape the batter onto the prepared baking sheet in an even layer. Bake for 20 to 25 minutes, until a toothpick inserted in the center comes out clean. Transfer the baking sheet to a rack to cool completely.
Meanwhile, make the icing In the bowl of the stand mixer fitted with the paddle, beat the butter at medium speed until fluffy, about 2 minutes. Beat in the crème fraîche just until incorporated. Add the confectioners’ sugar in two batches and beat at low speed just until blended. Beat in the vanilla. With the machine at high speed, gradually add the cream cheese by tablespoons until the icing is smooth.
Using the parchment, slide the cooled cake onto a work surface. Trim the cake edges to make them neat, then cut the cake crosswise into three even rectangles. Transfer one layer to a rectangular cake cardboard or cake plate. Spread about 1 cup of the icing evenly on top, then cover with another layer of the cake and frost with 1 cup more of icing. Set the last cake layer on top and frost the top and sides with the remaining icing. Refrigerate the cake until well chilled, at least 2 hours or overnight. Let the cake stand at room temperature for 1 hour before serving.