In a medium bowl whisk flour, baking powder, baking soda, cinnamon, ginger, cloves, nutmeg, salt, and brown sugar. With a rubber spatula stir in egg, applesauce, milk, and melted butter until combined.
Pour batter into a large zip-top plastic bag and cut one corner off. Pipe the batter into a 6-hole donut pan (I used a Wilton donut pan), filling half full. Bake for 8-9 minutes, or until lightly browned and cooked. Cool on cooling rack until donuts are cool enough to remove from pan.
To make the glaze:
Melt the butter and brown sugar in a small saucepan over medium heat. Bring to a boil and let cook for 2 minutes. Add 1 tablespoon milk and bring back to a boil. Add powdered sugar, using an additional tablespoon milk to thin out frosting if needed. Keep over low heat until ready to glaze donuts. Carefully dip donuts into glaze. Frosting will set quickly.