• Preheat oven to 350°.
• In a small bowl, mix together flour, baking soda, and salt; set aside.
• In a large mixing bowl, cream together butter and brown sugar. Add the egg and vanilla and beat until well combined. Add the peanut butter and mix to combine. Add the dry ingredients and mix until just blended.
• Form dough into 1 inch balls. Roll in sugar. Shape into a log. Roll log in sugar. Place logs on cookie sheet about 2 inches apart. Using the tines of a fork, make a criss-cross pattern in the dough. Pinch the middle of the cookie to make a peanut shape.
• Bake 10-12 minutes, until lightly browned on the edges and still soft in center. Allow cookies to cool on baking sheet for 5 minutes before removing to a wire rack to cool completely.

Filling

• In a large mixing bowl, beat butter on medium speed until fluffy. Beat in peanut butter, 1 cup powdered sugar, the milk and vanilla. Gradually beat in remaining powdered sugar until smooth.
• To assemble when the cookies have cooled completely, add a scant tablespoon of icing to the bottom side of one cookie and spread it out. Add a second cookie to the icing so that the two bottom sides are facing the frosting. Repeat with remaining cookies.