In a medium saucepan, melt the butter and the dark chocolate, stirring frequently, over medium heat. Let cool for 15 minutes.
While the chocolate is cooling, combine the flour, cocoa powder, baking powder and salt in a medium bowl and whisk together.
In the bowl of a stand mixer (or a large bowl if you are using a hand mixer), add the sugar, eggs and vanilla and mix on low speed until smooth, about 2 minutes.
Mix in the cooled chocolate mixture until just blended, then add the flour mixture in 2 batches, mixing on low speed until just combined. Cover the bowl with plastic wrap and refrigerate the dough for at least 1 hour.
Preheat oven to 325 degrees F. Line a large baking sheet with foil or parchment paper. Using a medium cookie scoop or tablespoon, scoop the dough and roll between your palms to form 1-inch balls. Arrange balls 2 inches apart on cookie sheet, and use a teaspoon or similarly sized/shaped item to make a thumbprint in each cookie. You may have to bake multiple batches depending on the size of your cookie sheet. Bake until the tops of the cookies crack, about 12 minutes.
Meanwhile, snip marshmallows in half crosswise. (Cut as you go, however many you need per sheet in the oven.) Remove the cookie sheets from the oven and spoon about a teaspoon of chocolate chips into the thumbprint of each cookie and top with one marshmallow half, pressing it down gently.
Return to oven and bake until the marshmallows are just softened, about 4 minutes. Transfer the pans to racks to cool for 5 minutes and grate the chocolate bar over the hot cookies. Transfer the cookies to racks to cool completely. Repeat the process with the remaining dough, marshmallows and chocolate.
Store in an airtight container, in single layers to prevent marshmallows from sticking to other cookies, for up to a week. And, I highly recommend popping in the microwave for a few seconds before eating any leftover cookies. They are delicious room temp, but that melty chocolate is so good!