Preheat your oven to 375 degrees. Wash, trim and chop one beet into large chunks. Wrap it in foil, place in oven and roast for about an hour, until very soft. Unwrap and let cool. (you can do this step even a few days ahead of time!)
To make the cake, preheat oven to 350 degrees. Butter a 9 x 13" pan and set aside.
Whisk together the flour, baking powder and salt. In another bowl, mix together the eggs, sugar, vanilla and oil.
Alternate adding the dry ingredients and the milk to the egg mixture, and mix until batter is well incorporated.
Pour batter into your prepared pan and bake for 40 minutes, until golden and sides are pulling away from the pan. Let cool completely.
To make the frosting, first start with the beet puree. Place the roasted beet in either a mini food processor or a blender (I used a blender) with about 1/4 cup water. Blend until very smooth. You don't want any chunks at all - make sure you get it really really smooth, even if it means adding more water. It will be fine.
To make the frosting, beat together the softened butter, sugar and vanilla until incorporated. Add as much beet puree as you want to get your achieved level of pink. I used about 2 tbsp to get a hot pink! Continue beating your frosting for about five minutes until very light and fluffy.
When cake as completely cooled, top with frosting. Enjoy!