Preheat oven to 425 degrees, Grease the donut pan.
In a medium bowl stir together flour, sugar, baking powder, cream of tarter, nutmeg, cinnamon, and salt.
Add in the milk, egg, and vanilla. Stir until just combined.
Gently stir in the diced peaches.
Spoon the batter into the donut pan. Alternatively, use a piping bag or zip top plastic bag to pipe the batter in. This helps to get the dough into the donut pan evenly and cleanly.
Bake for 8-10 minutes. Donuts should spring back to the touch.
Let cool in the pan for 5 minutes, then remove from pan onto a cooling rack. Cool completely before icing.
To Ice the Donuts
In a shallow dish combine the puréed half peach, vanilla extract, and 1 cup of powdered sugar. Stir until sugar is dissolved. Check the icing consistency. If you want it thicker, add more powdered sugar, if you need to thin it out puree the other half of the peach and use as needed, or use a tiny bit of milk or water.
Dip each cooled donut into the glaze to coat the top.