Line a cookie sheet with parchment paper and set aside.

Preheat oven to 350 degrees F.

Using a standing mixer or electric hand mixer, cream together butter, brown sugar and peanut butter until fluffy.

Add the egg and vanilla and continue mixing.

Add flour, baking soda, and salt and mix just until combined.

Divide dough into small balls, about 1 inch in diameter each then roll in the ⅓ cup sugar.

Place on cookie sheet about 2 inches apart.

Using the tines of a fork, make a criss-cross pattern.

Bake 10-11 minutes, until lightly browned on the bottom, but still soft in center.

Allow cookies to cool on baking sheet for a few minutes before removing to a wire rack to cool completely before assembling.

Using a stand mixer, beat butter on medium speed until fluffy.

Mix in peanut butter, 1 cup powdered sugar, milk, vanilla extract, and salt

Gradually mix in the remaining powdered sugar until smooth.

Spread about a tablespoon of frosting onto the flat side of one cookie, then press the flat side of another cookie on top to make a sandwich.