For the Peanut Butter Cookies:
Whisk together flour, baking soda and salt; set aside. Place butter, peanut butter and both sugars in the bowl of an electric mixer. Beat until light and fluffy. Add the egg and vanilla, beat to combine scraping down the sides of the bowl. With mixer on low, add the flour mixture, beat until incorporated. Wrap in plastic wrap or place in a bowl with a lid and refrigerate at least 30 minutes or overnight.
Preheat oven to 325°. Line two cookie sheets with parchment paper. On a slightly floured work surface roll out dough to 1/4-inch thickness. Using a sharp knife, cut dough into 2 1/2 X1-inch rectangles. Using the tines of a fork, score the cookies. Gather up scrapes and continue to make the rectangles until all the dough is used. Place cookies in the refrigerator about 15 minutes or until firm. Bake 15 to 20 minutes. Transfer to a wire rack to cool completely.
Using an offset spatula, spread 1 tablespoon of filling onto the flat sides of half of the cookies. Sandwich with remaining cookies, keeping the flat sides down. OR.....Spoon the filling into a piping bag and pipe on the filling.
For the Filling:
In bowl of an electric mixer, whip together the butter and peanut butter until light and fluffy. Add the powdered sugar and continue to beat until fluffy. Spread or pipe on cookies.