Preheat oven to 350°F (175°C). Line baking sheets with parchment paper.
In a bowl, whisk together flour, baking powder, baking soda, cinnamon, and salt.
In a separate bowl, beat butter and brown sugar until fluffy. Add egg, applesauce, and vanilla; mix well.
Gradually add dry ingredients to wet ingredients; mix until combined.
Drop spoonfuls of batter onto prepared baking sheets. Bake for 12-15 minutes, until edges are golden. Cool on racks.
For filling, beat butter until creamy. Gradually add powdered sugar, cinnamon, and vanilla. Add milk until desired consistency.
Spread filling on flat side of one cookie, top with another. Repeat with remaining cookies.