Quarter beets and transfer to food processor; process until pureed, scraping the sides down twice. (You should end up with a scant one cup of beet puree.)
Preheat oven to 350°F.
Line a 9-inch by 9-inch brownie pan with parchment paper or grease pan generously.
Cut sticks of butter into tablespoon size chunks and place in a heavy saucepan. Add chopped chocolate and cook over low heat, stirring constantly until mixture melts and is smooth. Remove the pan from heat and set aside to cool.
In a medium bowl, whisk together flour, baking powder and salt. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, gently mix the eggs for about 30 seconds. Add vanilla and brown sugar; mix on medium-high until light and airy, about 2 minutes. Reduce speed and add beet puree and then slowly add chocolate mixture; mix just until combined. Add flour mixture and again, mix just until combined. Pour batter into prepared pan and smooth top with a rubber spatula.
Bake 25 to 30 minutes, until a knife inserted into the center comes out with just a few moist crumbs sticking to it. Allow brownies to cool for 5 minutes then transfer with parchment to cooling rack.
Cut and serve warm, at room temperature, or straight from the refrigerator (my favorite).