Preparation

Make the shortcakes

• Combine the flour, cocoa, 1/4 cup of the sugar, the baking powder, baking soda, and salt in a food processor and pulse to combine. Crumble in the egg yolks, add the cubed butter, and pulse until the butter starts to become incorporated, about four 1-second pulses. Add the chocolate chips and continue to pulse until the butter pieces are no larger than peas, about four more 1-second pulses.
• Transfer to a large bowl, add the sour cream, and combine with a fork until the sour cream is mostly incorporated. Using your hand, gently fold and press the shaggy dough against the side of the bowl until all the dry bits are incorporated and it comes together in a cohesive but slightly tacky ball.
• Line a large rimmed baking sheet with parchment paper. Turn the dough out onto a lightly floured work surface and shape into a 1-inch-thick round, about 7-1/2 inches across. Using a 3-inch biscuit cutter, stamp out as many circles as possible and set them about 1 inch apart on the prepared baking sheet.
• Gently gather the scraps, press flat into a 1-inch-thick piece, and stamp out more rounds to yield a total of 6 shortcakes. Transfer them to the baking sheet and discard the remaining scraps. Freeze for 45 minutes.
• Position a rack in the center of the oven and heat the oven to 400°F. Brush the tops of the shortcakes with the melted butter, and sprinkle with the remaining 1 tsp. sugar. Bake until they’re mostly firm to the touch, 15 to 17 minutes. Cool for 3 minutes on the baking sheet, then transfer to a rack to cool completely.

Make the filling

• Using a serrated knife, split the shortcakes horizontally. In a large bowl, whisk the cream, sugar, and vanilla to medium-stiff peaks. Dollop a generous amount of the cream on the bottom halves, then top with some pomegranate seeds and the other shortcake halves. Dollop with more whipped cream on top, finish with more seeds, and serve.