Preheat oven to 350ºF. Line cookie sheets with parchment. Whisk flour, cinnamon, allspice, nutmeg, baking soda, baking powder, and salt. Set aside. Cream butter and sugar until light and fluffy. Beat in egg, applesauce, and vanilla bean paste until combined. Add the flour mixture and mix on low until combined. Scrape down bottom and sides of bowl. Portion onto prepared sheets using a 2-tablespoon cookie scoop. Bake for 12 minutes or until the tops bounce back when lightly pressed. Remove to a wire cooling rack. Once completely cooled, make the glaze. Whisk powdered sugar, milk, and vanilla bean paste. The mixture will seem thick at first, then it will loosen. Spoon or pipe over the cooled cookies.