Grease cake pan (sheet pan, 13×9 inch pan, or round pans), and preheat oven to 350 degrees F.
In bowl of stand mixer, beat sugar, butter, eggs, and vanilla for two minutes, scaping sides of bowl. Add applesauce and lemon juice, and beat two minutes more.
Combine flour, baking powder, baking soda, cinnamon, and salt, and whisk completely.
Mix into applesauce mixture alternately with buttermilk. Transfer to prepared pan(s) and bake as follows:
Sheet pan 30-40 minutes
13 x 9 inch pan 35-45 minutes
9 inch round pans 35-40 minutes
8 inch round pans 35-45 minutes
It’s done when a toothpick inserted into the center comes out clean. Cool completely.
Make the frosting by combining all the ingredients, and beating on high speed until light and fluffy. Spread on cake, drizzle with caramel, and top with walnuts if desired. Refrigerate two hours before serving. Refrigerate leftovers.