For the Cookies:
3 tbsp unsalted butter, softened
3/4 cup firmly packed brown sugar
1/2 cup unsweetened applesauce
1 large egg
1/3 cup dark molasses
3 cups all-purpose flour plus more for dusting
1 tsp baking soda
2 tsp ground ginger
1 tsp ground cinnamon
1/2 tsp allspice
1/4 tsp ground cloves

For the Icing:
1 1/2 tbsp egg whites, room temperature
1/4 tsp fresh lemon juice
1 cup powdered sugar
5 to 6 drops green food coloring
colored sprinkles