Preheat oven to 350F and line a baking sheet with parchment paper. Set aside.
In a small mixing bowl, whisk together flour, cocoa powder, baking powder, and salt. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment (or in a large mixing bowl with an electric hand mixer), beat together cream cheese and butter until smooth.
Add sugar to the cream cheese mixture, beating until well-combined.
Add eggs and vanilla extract, mixing well to incorporate them into the cream cheese mixture, scraping down the sides of the bowl as needed.
With the mixer on low speed, gradually add the dry ingredients to the bowl, mixing just until a soft cookie dough forms.
Use a cookie scoop to scoop out rounds of dough (alternatively, scoop out by rounded tablespoon), placing the balls of dough on the prepared baking sheet. If you want the cookies to have a smoother appearance, roll the dough balls in your hands, then place on the prepared baking sheet (the dough will be a little sticky--keeping your hands slightly wet or spritzed with nonstick spray will make it easier to roll the dough).
Bake in preheated oven for 10-12 minutes. Remove from oven and let the cookies cool on the baking sheet for about 5 minutes, then transfer to a wire rack to cool completely.
Prepare the icing by combining the chocolate chips, butter, and milk in a small microwave-safe bowl. Microwave for 30 seconds, then stir well. Continue microwaving in 15-second increments, stirring after each, until chocolate is fully melted.
Stir in 3/4 cup of sifted powdered sugar. If the mixture is too thick, add a couple more tablespoons of milk to thin it out. If the mixture is too thin, add a bit more powdered sugar.
Dip the tops of the cooled cookies in the icing, then place them on a piece of parchment or wax paper, icing side up, to allow the icing to cool and harden to a satiny finish.