Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan, and line the bottom with parchment paper for easier removal.
In a large mixing bowl, combine the flour, sugar, cocoa powder, baking powder, baking soda, and salt. Whisk until thoroughly mixed.
In another bowl, whisk together the cooked beet puree, vegetable oil, eggs, and vanilla extract until smooth.
Gradually add the wet ingredients to the dry ingredients, stirring gently until just combined. Do not overmix.
Pour the batter into the prepared cake pan and smooth the top. Bake in the preheated oven for 30 minutes, or until a toothpick inserted into the center comes out clean.
Once baked, remove the cake from the oven and allow it to cool in the pan for 10 minutes. Then, invert the cake onto a cooling rack and let it cool completely.
For the Beet-Vanilla Glaze, combine the powdered sugar, cooked beet puree, vanilla extract, and milk in a small saucepan. Heat gently while whisking until smooth. Adjust the milk to reach a pourable consistency.
Once the cake is completely cool, drizzle the Beet-Vanilla Glaze over the cake.