Preheat oven to 350°F and line baking sheets with parchment paper.
Beat butter, granulated sugar, and brown sugar together until light and fluffy.
Mix in egg, vanilla, and mashed banana. The mixture may look slightly separated at this stage, which is normal.
In a separate bowl, whisk together flour, baking soda, baking powder, salt, and cinnamon. Add to the wet ingredients and stir just until combined.
Scoop tablespoon-sized portions onto prepared baking sheets and bake for 10 to 12 minutes, until set and soft. Cool completely on a wire rack.
To make the frosting, beat butter and brown sugar together for 2 minutes until fluffy. Add powdered sugar, milk, vanilla, and salt and beat until smooth.
Spread frosting on the flat side of half the cookies and top with the remaining cookies to form sandwich cookies.