In a large mixing bowl, cream together softened unsalted butter and creamy peanut butter until smooth. Add granulated sugar and brown sugar, beating until fluffy. Mix in egg and vanilla until fully incorporated.

In a separate bowl, whisk together all-purpose flour, baking soda, and salt. Gradually add to the wet ingredients, mixing just until incorporated.

Preheat oven to 350°F (175°C) and line a baking sheet with parchment paper. Portion out dough onto the baking sheet, flatten slightly with a fork, and sprinkle with granulated sugar. Bake for 10-12 minutes until edges are golden. Cool completely.

Cream together softened unsalted butter, creamy peanut butter, and powdered sugar for the filling. Mix until smooth, adding milk gradually until desired consistency is achieved.

Once cookies are cooled, pair them up. Spread peanut butter filling on one cookie and press another on top to form a sandwich. Repeat with remaining cookies and filling.