Preheat oven to 425 degrees. Spray and flour donut pans thoroughly.
In medium bowl, whisk flour, baking powder, salt, cinnamon and nutmeg together. Set aside.
In large bowl, whisk egg, milk, vanilla and melted butter together. Stir in carrots. Fold wet ingredients into dry ingredients, just until combined. Spoon batter into donut pan, filling about half to three-fourths full.
Bake in 425 degree oven for 7-9 minutes or until donuts spring back to the touch and are golden on edges. Let sit in pan for a minute, then turn out onto a cooling rack. Let cool completely.
To make glaze: Beat together cream cheese, powdered sugar, vanilla and milk and drizzle over cooled donuts.