Preheat oven to 350ºF. Line the bottom of an 8 x 4-inch loaf pan with parchment paper. Grease and flour the insides.
In a medium-sized bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon and nutmeg until well blended.
Set aside 2 tablespoons of the peach purée for the icing.
In a large bowl, beat the sugar, oil, eggs, remaining peach purée and the extracts with an electric mixer at medium speed or by hand with a whisk, until smooth and creamy. Add the flour mixture and stir just until combined.
Transfer the batter to the prepared baking pan. Bake until a toothpick inserted into center comes out clean, about 38-45 minutes.
Let the cake cool in pan for 10 minutes, then turn it out onto a cooling rack to cool completely. Spread the icing on the cooled cake.