Preheat the oven to 375 degrees F. Line two baking sheets with parchment paper.
Melt the chocolate and 1 stick of the butter over medium heat in a double boiler. Remove from the heat and set aside to cool slightly.
In a medium bowl, whisk together the flour, cocoa powder and baking powder. Set aside.
To the bowl with the melted chocolate, add the brown sugar, granulated sugar, vanilla and salt. Whisk to combine. Whisk in the eggs, one at a time, followed by the yogurt. Using a rubber spatula, fold in the flour mixture being careful to avoid over-mixing.
Scoop 1-tablespoon mounds of the batter on the prepared baking sheets, leaving about 1-inch between each cookie. Bake the cookies for about 9 minutes, rotating the trays halfway through. Slide the cookies on the parchment onto a cooling rack and cool completely.
Meanwhile make the filling. In a medium bowl, use an electric hand mixer to beat the remaining 2 sticks butter and the confectioners' sugar; mix until light and fluffy, starting on low speed to avoid splatter and work up to medium speed, 3 minutes. Add the peppermint extract, salt and peppermint candies. Beat on medium speed until the candies are broken apart and evenly distributed, 2 minutes.
Spread the filling on the flat side of half of the cookies. Top with the remaining cookies. Roll the edges in the crushed candy canes if using and serve immediately or store in an airtight container until ready to serve.