In a medium saucepan, melt the butter and chopped chocolate, stirring frequently, over medium heat. Let cool for 15 minutes.
In a medium bowl, whisk together the flour, cocoa powder, baking powder and salt.
Using an electric mixer, beat the sugar, eggs and vanilla at low speed until smooth, 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; preheat to 325 degrees . Line 2 large cookie sheets with parchment. Using a tablespoon, scoop the dough and roll between your palms to form 1-inch balls. Arrange about 16 balls 2 inches apart on each cookie sheet, flattening slightly. Bake until the tops of the cookies crack, about 12 minutes.
Meanwhile, snip 8 marshmallows in half crosswise and stick 1 square of chocolate onto each of the cut sides.
Remove the cookie sheets from the oven; gently press a marshmallow half, chocolate side down, into each cookie. Bake until the marshmallows are just softened, about 4 minutes. Transfer the pans to racks to cool for 5 minutes; grate the remaining chocolate over the hot cookies. Using a spatula, transfer the cookies to the racks; let cool. Repeat the process with the remaining dough, marshmallows and chocolate. Bake each batch on a clean sheet of parchment.