Preheat oven to 350F with rack on lower middle position. Line baking sheet with parchment paper.
In a large bowl, whisk together flour, salt, baking soda, and cocoa.
In another large bowl or the bowl of stand mixer, combine butter, both sugars, and vanilla. Beat on medium-high until it's as fluffy as it can get, scraping sides down periodically; about 3-4 minutes. Add beaten eggs and beat just to combine.
Using rubber spatula, gently fold flour mixture into butter mixture just until batter is completely moistened. Fold in milk chocolate chips.
Wrap and chill dough until firm enough to handle, about 30 min to 1 hour.
Form dough into walnut sized balls and flatten slightly, as these tend not to spread too much. Place on parchment lined baking sheet and bake about 8 minutes. They'll seem a bit underdone, but will set up nicely when cool.
Meanwhile, make the filling: Beat together the butter and peanut butter in a large bowl until smooth and fluffy. Gradually add in powdered sugar and beat until filling is smooth. Add 1 TB milk at a time, until desired consistency. Gently fill cooled cookies with desired amount.