Prepare a springform pan and line the bottom with parchment paper.
Boil the beets in salted water until they’re very tender (when you stick a knife in them it should just slide right in) or about 45 minutes. Drain then rinse the beets with cold water. When cool enough to handle, slip off the peels.
Roughly chop the beets and then put them in a food processor until you get a coarse purée. (If you don’t have a food processor, use a cheese grater.)
Preheat the oven to 350ºF.
In a double broiler on the stove, 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 water and stir it once.
Then add the butter and gently stir a few times.
Sift together the flour, cocoa powder, and baking powder in a separate bowl.
Remove the bowl of chocolate from the heat and stir until the butter is completely melted. Let sit for a few minutes to cool.
Stir the egg yolks together and briskly stir them into the melted chocolate mixture.
Add in the vanilla and fold in the beets.
In a stand mixer whip the egg whites until stiff. (about 3 minutes)
Remove bowl from mixer and gradually fold the sugar into the whipped egg whites with a spatula.
Once incorporated, fold the egg whites into the melted chocolate mixture, being careful not to overmix.
Fold in the flour and cocoa powder.
Scrape the batter into the prepared cake pan and REDUCE THE HEAT of the oven to 325ºF 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.
Sift powdered sugar evenly over the cake for serving.