Preheat oven to 375 degrees F. Grease cookie sheets; set aside. In a large bowl beat butter with an electric mixer on medium to high speed for 30 seconds. Add brown sugar, cinnamon, baking powder, and salt. Beat until combined and fluffy, scraping sides of bowl occasionally. Beat in egg and vanilla until combined. Beat in as much of the flour as you can with the mixer. Stir in any remaining flour.
Shape dough into 1-inch balls. Place 1 1/2 inches apart on prepared cookie sheet. Dip the bottom of a drinking glass into granulated sugar. Flatten dough balls to circles about 2 inches in diameter by gently pressing with the glass bottom.
Bake in the preheated oven about 8 minutes or until edges are light brown. Cool on cookie sheet for 2 minutes. Transfer to a wire rack; cool completely.
Spread the Cinnamon Filling on the flat sides of half of the cookies. Top with the remaining cookies, flat sides down. If desired, roll edges in white sprinkles.
In a medium bowl beat butter, softened; vanilla; and ground cinnamon with an electric mixer on medium to high speed. Gradually add 1 cup powdered sugar, scraping sides of bowl occasionally. Beat in milk or apple cider. Gradually add 1 cup additional powdered sugar. Add additional milk or apple cider, if needed, to make a filling of spreading consistency.