1. Preheat oven to 350°F with rack in middle. Lightly butter a 9-inch springform pan.
2. In a small bowl, whisk together flour, baking powder, and salt.
3. In a large bowl, beat butter and 3/4 cup sugar with an electric mixer until pale and fluffy. Add eggs 1 at a time, beating well after each addition, then beat in extracts. At low speed, mix in flour mixture until just combined.
4. Spread batter evenly in pan, then scatter peaches over top. Stir together nutmeg and remaining 1/2 tablespoon sugar and sprinkle over top.
5. Bake until cake is golden-brown and top is firm but tender when lightly touched (cake will rise over fruit), 45 to 50 minutes. Cool in pan 10 minutes. Remove side of pan and cool to warm.