Preheat the oven to 350F. Line the bottom of an 8×4-inch loaf pan with parchment; grease and flour the sides.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, and nutmeg.
Set aside 2 tablespoons peach puree for the icing.
In a large bowl, beat the sugar, oil, eggs, remaining peach puree, and extracts until smooth.
Add the flour mixture and beat just until combined.
Transfer the batter to the prepared pan. Bake until a toothpick inserted in the center comes out clean, about 38-45 minutes.
Cool in the pan 10 minutes, then turn out onto a rack to cool completely.
For the icing, beat together the cream cheese, butter, powdered sugar, vanilla, salt, and reserved peach puree until smooth. Spread over the cooled cake and top with diced peaches.