For the cookies: Into a medium-sized microwave safe bowl add the butter and semi-sweet chocolate chips. Heat it in the microwave, stirring every 30 seconds, and continuing for additional 30 second heat bursts until melted. Then whisk in the sugar until well combined. Then whisk in the vanilla. Then whisk in the egg until combined. Set aside.
In another bowl, whisk flour, cocoa, salt and baking soda until combined. Stir flour mixture into the chocolate mixture. Beat with a mixer until the dough comes together.
Transfer the dough to a sheet of wax paper. Shape into a log about 2 1/2 inches in diameter. Place log along the edge of the wax paper and roll it up in the paper. Shape it as you roll to maintain a smooth cylinder. Twist the ends of the paper to secure it closed and help keep the shape compact.
Refrigerate at least two hours or until firm. If log settles and flattens on one side, rotate every 15 minutes or so to maintain a round shape.
Heat oven to 325 F and line baking sheet with parchment paper.
Unwrap the dough and put it onto a cutting board. Using a sharp knife, cut dough into quarter inch thick slices and place on prepared baking sheet. Bake for about 12-18 minutes or until cookies are firm to the touch. Just watch them because it’s hard to tell by the looks if they are done since they are so dark to begin with. These bake in batches well, so bake a few to test the bake time, if you have the time. And if your cookies are smaller, adjust the baking time.
When done remove the pan from the oven. Allow cookies to cool while you prepare the filling.
In the bowl of a stand mixer, beat the butter until smooth and creamy. Add sugar and vanilla and beat until smooth. Add milk and salt and continue beating until smooth again. Place about tablespoon of filling on the bottom of one cookie and about a tablespoon of peanut butter on another cookie. Sandwich the two cookies together. Repeat with remaining cookies.