Make the cookies: 

1. In a large mixing bowl, mix together the melted butter, sugar and peanut butter using an electric mixer until smooth and completely incorporated. Add in the eggs and vanilla extract one at a time, mixing well between each addition.
2. In a separate bowl, whisk together the flour, cornstarch, baking soda and baking powder. Add the dry ingredients to the wet ingredients and mix just until there are no dry streaks remaining. The dough should be thick.  
3. Cover and refrigerate the dough for at least one hour. 
4. Preheat the oven to 325˚F and line two baking sheets with parchment paper.
5. Scoop the dough out in one teaspoon-sized portions. Roll each portion into a ball, then place two balls next to each other (touching) on the baking sheet, spacing them about 1 1/2 inches apart. (It's quickest to scoop all of the dough first, then roll into balls after.)
6. Using the tines of a fork, press down widthwise and then lengthwise to create a criss cross pattern and to flatten out the dough. 
7. Bake for 13-15 minutes, until the bottoms are deep golden brown. Remove from the oven and let cool for a few minutes, then transfer the cookies to a cooling rack to cool completely before filling. 

Make the filling: 

1. Add the peanut butter, powdered sugar, 4 tablespoons of milk and vanilla to a mixing bowl and mix until smooth and completely incorporated. If it's too thick to pipe, add an additional 1 tbsp of milk. 
2. Transfer the mixture to a piping bag (or just any regular plastic bag and cut off the corner to pipe). Flip over a cookie so the flat bottom is facing up. Pipe a ~1/2 teaspoon dollop on each side of the cookie. Place another cookie on top to create a sandwich. Repeat and enjoy!