Add the butter and semisweet chocolate chips to a medium saucepan over medium heat.
Stir until completely melted. Let cool for 15 minutes.
In a medium bowl, whisk the flour, cocoa powder, baking powder, and salt. Set aside.
Beat the sugar, eggs and vanilla at low speed until smooth, about 2 minutes.
Mix in the cooled chocolate mixture just until blended.
Add the flour mixture in 2 batches, mixing on low speed until just combined.
Refrigerate the dough for at least 1 hour. (If making the dough a day ahead, let sit at room temperature for 30 minutes before shaping.) Position a rack in the upper third of the oven and preheat the oven to 325 degrees. Line 2 large cookie sheets with parchment paper.
Using a tablespoon, scoop the dough and roll between your palms to form 1-inch balls. Arrange about 12 balls 2 inches apart on each cookie sheet, flattening slightly.
Bake until the tops of the cookies crack, about 12 minutes.
Remove the cookie sheets from the oven; gently press 2 to 3 mini marshmallows into each cookie. Press 2 to 3 milk chocolate chips into each cookie.
Return the cookies to the oven and bake until the marshmallows are just softened, about 4 minutes.
Grate the dark chocolate over the hot cookies. Cool the cookies on the sheets for 5 minutes, then transfer the cookies to racks to let cool completely.
Repeat with the remaining dough. You should end up with about 3 dozen cookies.