Preheat oven to 350 degrees.
In a large mixing bowl, add in butter, applesauce, sugars, and eggs and beat over medium speed until combined.
Add in pumpkin and continuing beating until thoroughly incorporated.
In a liquid measuring cup, combine milk and lemon juice and set aside.
In a medium bowl, combine remaining dry ingredients.
Slowly, over medium speed, add the dry ingredients to the pumpkin mixture, alternating with the milk mixture, beating well after each addition.
Line cupcake trays with paper liners.
Fill cupcake liners 3/4 of the way full and bake for 24 minutes or until a skewer inserted into the middle comes out clean. Remove from oven and allow to cool.
To make frosting, combine all ingredients in a medium mixing bowl and beat over medium speed until ingredients begin to combine, slowly increase speed continue beating until a frosting is formed. Once cupcakes cool completely, frost as desired and enjoy!