Cover a baking sheet with parchment paper and set aside.
Cook the beets in boiling water until tender, approximately 15 minutes. Test a beet by poking it with a fork — if it comes out easily, the beet's done. Peel the beets while they are still warm. Roughly chop them and puree in a blender with the buttermilk, until smooth. Cool completely.
In a large mixing bowl, combine the butter with the sugars. Add the cooled beet puree, cocoa powder, vanilla, vinegar, salt and egg, and mix to blend. Add the flour and baking soda and mix as little as possible to fully incorporate them. Fold in the white chocolate chips.
Using a 1 ¼-inch ice-cream scoop or a teaspoon, form 1 ¼-inch balls of dough and place them on the parchment-covered baking sheet. Cover with plastic wrap and refrigerate for at least 2 hours. (This is a sticky dough, so working with your hands isn't easy.)
When you are ready to bake the cookies, cover another baking sheet with parchment paper and place about 12 dough balls on each sheet — they will spread a bit. Bake at 350 for approximately 10 to 12 minutes. Cool the cookies on a cooling rack for at least 10 minutes before serving.
Makes approximately 5 ½ dozen cookies.