1. Make cake: Preheat oven to 350ºF. Grease two 9-inch round pans and line with parchment.
2. Whisk buttermilk, oil, eggs, sugar, applesauce and molasses in a large bowl until smooth. Sift dry ingredients on a sheet of waxed paper. Add to buttermilk mixture, stirring to combine.
3. Divide batter between pans and bake in center of oven until a toothpick inserted in center comes out clean, about 20 to 25 minutes. Let cool in pan for 10 minutes, then invert onto rack to cool completely. (Can be made one day ahead. Wrap in plastic and store at room temperature.)
4. Make frosting: In a bowl with electric mixer, beat butter until smooth and creamy. Slowly add sugar, beating constantly. Add syrup and extract (if using), beating until well combined.
5. Assemble cake: Reserve 1/4 cup of frosting. Place one cake layer on serving plate. Spread with 1 generous cup of frosting; top with second cake layer. Spread remaining frosting on top and sides of cake. Chill for 2 hours or overnight.
6. If cake was refrigerated overnight, bring to room temperature. Spread a dollop of reserved frosting on store-bought gingerbread cookies to stick them to cake (or make your own cookies).