Position the racks in the lower and upper thirds of the oven and preheat the oven to 325°F. Line a 17 by 12 by 1-inch baking sheet (half sheet pan) with a Silpat or parchment paper.

In a small bowl, combine the flour, baking powder, and salt, and set aside.

In the bowl of a stand mixer fitted with the paddle attachment, combine the butter, brown sugar, and granulated sugar and mix on medium speed to blend, 1 to 2 minutes. Add the vanilla, egg, and peanut butter and mix for 2 to 3 minutes. Pour in the melted chocolate and mix until just combined. Stop the mixer, add the dry ingredients, and mix until just combined. Remove the bowl from the mixer and fold in the chocolate pieces.

Drop large mounds of dough, about 3 heaping tablespoons each, onto the prepared baking sheet, leaving about 2 inches between them. Bake for about 20 minutes, or until the cookies are lightly cracked on top, rotating the pans halfway through. Remove from the oven and cool on the sheets for 2 minutes, then transfer to a cooling rack to cool completely.