Heat oven to 400 degrees. Slice peaches; sprinkle with kirsch or schnapps, if desired, and 1 tablespoon granulated sugar. Set aside.
In the bowl of a food processor, combine the rest of the dry ingredients. Add butter; process with on-off pulses until mixture resembles coarse meal. While pulsing, slowly add enough cream or milk until dough just comes together.
Turn out dough onto a lightly floured board, and knead once or twice. Pat into a 3/4-inch-thick square. Trim edges, and cut into four squares. Brush with additional cream or milk; sprinkle with sugar.
Place on a lightly buttered baking sheet; bake for 20 to 25 minutes. Transfer cakes to a wire rack, and allow to cool slightly.
Meanwhile, whip the cream: In a large bowl, beat the heavy cream and vanilla extract until stiff peaks form; set aside. Split cakes horizontally. Place bottom halves on dessert plates, and top with peaches and vanilla whipped cream. Cover with top halves of shortcakes, and serve.