stick unsalted butter, melted and cooled slightly plus more for buttering the dish
large egg, at room temperature
milk, at room temperature
plus 1/4 cup granulated sugar
peaches, peeled and cut into wedges
pure vanilla extract
1. Preheat the oven to 375°. Generously butter a 9-by-13-inch glass baking dish. In a large bowl, using an electric mixer, beat the eggs with the milk until frothy. Add the 1 1/3 cups of granulated sugar and beat at high speed until the mixture is thick and pale, about 7 minutes.
2. In a medium bowl, sift the flour with the baking powder and salt. Fold the dry ingredients into the beaten egg mixture, then fold in the melted butter until incorporated. Spread the batter in the prepared dish and bake in the center of the oven for about 30 minutes, until the cake is golden. Transfer to a rack and let cool.
3. Meanwhile, in a medium bowl, mix the remaining 1/4 cup of granulated sugar with the peach schnapps. Stir in the peaches and let stand at room temperature for at least 15 minutes, stirring occasionally.
4. In a bowl, whip the cream with the confectioners' sugar and vanilla until firm. Cut the shortcake into squares and serve with the peaches and whipped cream.