In a medium saucepan, bring the beets and 2 1/4 cups water to a simmer over medium heat; cover and cook until fork-tender, about 35 minutes. Transfer the beets with a slotted spoon to a food processor, reserving the cooking water. Puree the beets. Add the reserved cooking water as necessary to get 2 1/4 cups beet puree; transfer to a medium bowl. Whisk in the brown sugar and let cool.
Meanwhile, position a rack in the lower third of the oven and preheat to 350 degrees . Line a dozen 1-cup muffin cups with jumbo baking liners. In a medium bowl, whisk together the flour, cocoa, baking powder, baking soda and 3/4 teaspoon salt.
Whisk the eggs into the beet puree. Whisk in the melted butter and yogurt until smooth. Add the dry ingredients in 2 additions, stirring with the whisk until smooth. Spoon the batter into the cupcake cups. Bake until springy and a toothpick inserted in the center comes out clean, about 35 minutes. Let the cupcakes cool for at least 30 minutes.
Make the frosting. Using an electric mixer, beat the softened butter and the cream cheese until fluffy. Mix in the confectioners sugar and pinch salt at low speed. Beat in the white chocolate and vanilla at medium speed until fluffy.
Using a pastry bag (or a resealable plastic bag with the corner snipped), pipe the frosting generously onto the cupcakes, starting from the center and going out to the edge, then returning to center. Sprinkle with the sanding sugar.
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