Heat oven to 375°F. In the bowl of a stand mixer fitted with the paddle attachment, beat together the butter, sugar, salt, peppermint extract, and melted chocolate until creamy, about 2 minutes. Reduce speed to low.
Add flour and beat until mixture forms a dough, about 2 minutes. Stir in 1 cup mini chocolate chips by hand.
Using a tablespoon, mold dough into a 1 ½ inch ball. With your hand, flatten balls into a 1-1/2-inch round and place on prepared baking sheet. Bake for 5 to 7 minutes or until set. Cool for 2 minutes on the baking sheet before transferring cookies to a wire rack to cool completely.
To make the frosting: In the bowl of a stand mixer fitted with the paddle attachment, beat together all frosting ingredients.
Add milk, 1 teaspoonful at a time, until frosting is smooth. Frost cooled cookies. Place on waxed paper.
To make the drizzle: Place chocolate chips and shortening in small microwave-safe bowl. Microwave on HIGH, stirring every 30 seconds, until smooth. Drizzle thin lines of chocolate over frosting.
Let stand until chocolate is set (at least 30 minutes).