In a large bowl, cream the butter and sugar together for 5-7 minutes until the mixture is light and fluffy.
Beat in the sliced banana and vanilla extract until fully incorporated.
In another bowl, whisk together the flour and salt. Gradually add the dry mixture to the creamed ingredients, mixing well.
Stir in the chopped salted peanuts for added crunch.
Divide the dough into two portions and shape each into a disk. Wrap in plastic and refrigerate for 30 minutes to firm up.
Preheat your oven to 350°F. On a lightly floured surface, roll out each disk of dough to a 1/4-inch thickness.
Use a 2-inch cookie cutter to cut out rounds and place them 1 inch apart on ungreased baking sheets.
Bake for 10-12 minutes or until the edges are lightly golden. Allow the cookies to cool completely on wire racks.
For the frosting, beat together the confectioners' sugar, peanut butter, softened butter, vanilla extract, and enough milk to achieve a creamy consistency.
Drop a heaping teaspoon of frosting onto the center of half the cookies. Press the remaining cookies gently on top to spread the filling evenly.