In a medium saucepan, bring the beets and the water to a simmer over medium heat. Cover and cook until fork tender, about 35 minutes.

When tender, using a slotted spoon transfer the beets to a food processor or blender. Reserve the cooking water. Puree the beets, and add enough of the cooking water to make 2 1/4 cups of puree. Transfer to a medium sized bowl. Stir in the brown sugar and set aside.

Place the oven rack in the bottom third of the oven and preheat the oven to 350°F. Line muffin tins with baking liners.

In a medium bowl, whisk together the flour, cocoa powder, baking soda, and 3/4 teaspoon of the salt.

Whisk the eggs into the beet puree. Whisk in the melted butter and the yogurt. Add the dry ingredients in 2 batches, making sure to whisk until smooth.

Spoon the mixture into the baking cups, filling 2/3 full. Bake in the preheated oven until a toothpick comes out clean, about 27 to 30 minutes. Let cool completely.

When the cupcakes are cool, make the frosting: Using an electric mixer, beat the softened butter and the cream cheese until smooth. Mix in the confectioners' sugar and a pinch of salt. Beat in the white chocolate and vanilla until fluffy.

Using a pastry bag, pipe the frosting onto the cupcakes.

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Adapted from rachaelray.com