How to Make It
Preheat the oven to 425°. Place the flour, cornmeal, baking powder, salt and 3 heaping tablespoons of sugar in a food processor. Pulse to combine. Add the butter and pulse 10 times, to a coarse meal. With the machine on, quickly pour in 1 cup of the cream. Stop the machine immediately when the dough starts to come together. (It is important not to overwork the dough.)
Place the dough on a clean work surface and bring it together with your hands. Shape it into an even 8-inch round. Cut the round in half, then cut each half into 3 wedges. Place the shortcakes on a buttered baking sheet. Brush them with the remaining tablespoon of cream and sprinkle a little sugar on top of each one. Bake about 15 minutes, until the shortcakes are set and lightly golden.
Cut the shortcakes in half horizontally. Place the bottom halves on each of 6 plates or on a large platter. Spoon some of the peach puree onto the shortcakes and top with a large dollop of the soured cream. Ladle the peaches and their juices over the cream, letting some of the fruit fall onto the plates. Drizzle on a little more peach puree, set the shortcake tops in place and serve.