Preheat oven to 350ºF. Line the bottom of an 8 x 4-inch loaf pan with parchment paper. Grease and flour the sides.
In a medium bowl, whisk together flour, baking powder, baking soda, salt, cinnamon and nutmeg.
Set aside 2 tablespoons peach purée for icing.
In a large bowl, beat sugar, oil, eggs, remaining peach purée and extracts until smooth.
Add flour mixture and beat just until combined.
Transfer batter to prepared baking pan. Bake until toothpick inserted into center comes out clean, about 38-45 minutes.
Let cake cool in pan for 10 minutes, then turn out onto a cooling rack to cool completely. Spread icing on cooled cake.
Icing
In a medium bowl, beat cream cheese and butter together until smooth.
Add powdered sugar, salt, vanilla extract and 1 tablespoon of peach purée. Beat until smooth. Add additional peach purée, if needed, until desired consistency is reached. Fold in diced peaches.