Preheat the oven to 350 degrees F. Line baking sheets with parchment paper and set aside.
In a mixing bowl, combine the cake mix, egg and butter. Beat with a mixer on low speed until all the cake mix has been fully incorporated. Don't over mix the dough.
Roll the dough into 1-inch balls and place on baking sheets. Try to make them all the same size so you may have to do some readjusting with the size of the balls until they are all the same size. You should get about 24 balls to make 12 sandwiches.
Slightly flatten the dough before it goes into the oven because they rise a lot.
Bake for 8-10 minutes in the oven. Allow cookies to cool completely before icing.
In a mixing bowl, combine the softened butter with the powdered sugar and mix with a blender on low speed until they are well blended. A
dd the vanilla extract and one teaspoon of the milk and mix well. Add milk one teaspoon at a time until the frosting is the desired consistency. Make sure that you scrape the sides of the bowl before adding more milk each time.
To assemble the sandwiches, frost the flat side of one cookie with the icing. With another cookie, make a sandwich and press down so that the icing reaches the sides of the cookie so that there is something for the sprinkles to adhere to.
Place the sprinkles in a bowl and roll the edges of the cookie in the sprinkles to cover.