Preheat the oven to 325°F with a rack in the center position. Line two sheet pans with parchment paper.
Make the cookies. In a medium bowl, whisk together the flour, cocoa, baking powder, baking soda, and salt. In a large bowl, whisk together the brown sugar, eggs, and vanilla.
Place the butter and chocolate in a small, microwave-safe bowl. Microwave 10 seconds at a time, whisking in between, until melted. Whisk the chocolate mixture into the egg mixture, then fold in the dry ingredients. Mix until no dry parts remain. Let sit at room temperature for about 10 minutes, then form into 1½-inch balls.
Place the granulated sugar and powdered sugar in separate shallow bowls. Working with 1 ball of dough at a time, roll the ball in the granulated sugar until coated. Transfer the dough ball to the powdered sugar and roll until coated. Evenly space the dough balls on the prepared sheet pans.
Working with one sheet pan at a time, bake the cookies, rotating halfway through, until puffed and soft in the center and cracks have formed around the edges, about 12 minutes. Let cool on the sheet pan completely.
Meanwhile, make the frosting. In the bowl of a stand mixer fitted with the paddle attachment, beat the butter on medium until smooth and fluffy. With the mixer on low, gradually add the powdered sugar and beat until incorporated. Add the cream and peppermint extract and mix until smooth.
Place the crushed candy canes on a plate. Once cookies have cooled completely, spread about 2 tablespoons of frosting on the bottom of one cookie and press it to the bottom of another cookie to create a sandwich. Roll the edges of the frosting in the crushed candy canes. Repeat with the remaining ingredients.
