In a medium bowl, whisk together flour, salt, and baking soda. In a large bowl with a mixer, beat softened butter and peanut butter together until creamy. Add in both sugars, beat until fluffy.
Add in egg and vanilla extract, beat until smooth. Add the flour mixture in three parts, beating well after each until fully combined. Refrigerate dough for at least one hour. Preheat oven to 350 F. Roll cookie dough into logs, place 2-inches apart on ungreased baking sheets.
Flatten slightly and press with a fork in a criss cross pattern, pinch the middles together. Bake for 10-12 minutes. Let cool completely.
In a large bowl with a mixer, beat softened butter with peanut butter until creamy. Add in powdered sugar, milk, and vanilla extract, beating until smooth. Place the peanut butter filling onto half of the peanut butter cookies. Sandwich with another cookie.
