Preheat oven to 325°F and line two baking sheets with parchment paper or silicone liners.
In a small saucepan over medium-low heat, melt 4 tablespoons of butter and chocolate chips until completely smooth. Set aside to cool for 10 minutes.
Meanwhile, in a large bowl whisk brown sugar and eggs. Set aside.
In a separate bowl, whisk flour, cocoa powder, baking powder, baking soda, and salt.
Pour the melted chocolate mixture into the eggs and brown sugar and whisk until combined. Add the flour mixture and stir until combined (I stir it with a rubber spatula). The dough will be sticky, but if it seems too wet to scoop, mix in another tablespoon or two of flour.
Scoop dough by the tablespoon (I use a scoop) and place 2 inches apart on prepared baking sheets. Dampen your palm, and lightly press down on each mound of dough to flatten slightly. Bake for 9-10 minutes, until the edges are set and the centers are puffed and still appear slightly underdone (they will continue to cook as they cool). Place baking sheets on cooling racks and cool completely.
Filling:
Using a hand mixer or stand mixer, beat butter until smooth. Add powdered sugar and beat until combined. Add milk and peppermint extract and beat on medium speed for 1 minute, until smooth and creamy.
Using a butter knife or pastry bag, spread or pipe frosting onto the bottom of half the cookies. Top with the remaining cookies. If desired, roll the edges (where the frosting is) in crushed candy canes.
Notes
I made these cookies with 1 ½ tablespoons of dough, which made a dozen 3-inch cookies. If you'd like your cookies a little smaller, feel free to do so. Just adjust the baking time accordingly, reducing it by several minutes. Be sure not to overbake; the cookies should be soft.