For the cupcakes:
½ cup plus 1 tablespoon Dutch-process cocoa powder
½ cup plus 1 tablespoon hot water
2¼ cups all-purpose flour
¾ teaspoon baking soda
¾ teaspoon baking powder
½ teaspoon salt
2 sticks plus 1 tablespoon unsalted butter, at room temperature
1⅔ cups sugar
3 large eggs, at room temperature
1 tablespoon vanilla extract
¾ cup sour cream
For the frosting:
4 ounces unsweetened chocolate, melted and cooled
3 cups powdered sugar (sifted if lumpy)
1 cup (2 sticks) unsalted butter, at room temperature
Pinch of fine sea salt
2 tablespoons heavy cream or whole milk
1 teaspoon vanilla extract
Heat oven to 350º. Line cupcake pans with paper liners. Combine the cocoa powder and water in a small bowl and whisk until smooth. Combine the flour, baking soda, baking powder and salt in a medium bowl and whisk to combine.
Combine the butter and sugar in a medium saucepan and place over medium heat. Cook, stirring occasionally, until the butter is completely melted and the mixture is smooth. Transfer to the bowl of an electric mixer and beat on medium-low speed for 4-5 minutes or until the mixture is cool. Beat in the eggs one at a time, mixing well after each addition. Blend in the vanilla and cocoa powder and continue mixing until smooth, scraping down the sides of the bowl as needed.
With the mixer on low speed, blend in the flour mixture in three additions, alternating with the sour cream, beginning and ending with the dry ingredients. Mix each addition until just incorporated, then scrape down the sides and bottom of the bowl. Divide the batter evenly between the prepared liners and bake until a toothpick inserted in the center comes out clean, 18-20 minutes, rotating the pans halfway through baking. Cool in the pans for at least 5 minutes, then transfer to a cooling rack to cool completely before frosting.
While the cupcakes cool, make the frosting. Combine all the ingredients in the bowl of a food processor and pulse to combine. Scrape down the bowl and continue processing until the frosting is smooth and fluffy. You can certainly do this with a hand mixer or stand mixer, as well. Just beat the butter and sugar together first, scrape down the bowl, then add the remaining ingredients and whip until fluffy.