1. Butter an 8 or 8 1/2 inch (20 cm) springform pan and line the bottom with parchment paper.
2. Boil the beets in salted water with the lid askew until they’re very tender when you stick a knife in them about 45 minutes. Drain then rinse the beets with cold water. When cool enough to handle, slip off the peels, cut the beets into chunks, and grind them in a food processor until you get a coarse, yet cohesive, puree. (If you don’t have a food processor, use a cheese grater.)
3. Preheat the oven to 350ºF (180ºC).
4.In a large bowl set over a pan of barely simmering water, melt the chocolate, stirring as little as possible. Once it’s nearly all melted, turn off the heat (but leave the bowl over the warm water), pour in the hot espresso and stir it once. Then add the butter. Press the butter pieces into the chocolate and allow them to soften without stirring.
5. Sift together the flour, cocoa powder, and baking powder in a separate bowl.
6. Remove the bowl of chocolate from the heat and stir until the butter is melted. Let sit for a few minutes to cool, then stir the egg yolks together and briskly stir them into the melted chocolate mixture. Fold in the beets.
7. In a stand mixer, or by hand, whip the egg whites until stiff. Gradually fold the sugar into the whipped egg whites with a spatula, then fold them into the melted chocolate mixture, being careful not to overmix.
8. Fold in the flour and cocoa powder.
9. Scrape the batter into the prepared cake pan and reduce the heat of the oven to 325ºF (160ºC), and bake the cake for 40 minutes, or until the sides are just set but the center is still is just a bit wobbly. Do not overbake.
Let cake cool completely, then remove it from the pan.