Preheat the oven to 350 degrees Fahrenheit. Butter an 8- or 9-inch square baking pan (I used 8-inch only because I didn't have a 9-inch). Set aside.
Sift together the flour, cocoa powder, and salt; set aside.
Melt half the butter in a medium bowl in the microwave. Add the remaining butter to the melted butter and stir. The butter should look creamy, with small bits of unmelted butter, and be at room temperature.
In a bowl of a stand mixer fitted with the paddle, mix together the eggs and sugar on medium speed for about 3 minutes, or until thick and very pale. Mix in the vanilla and peppermint extracts. On low speed, add about one-third of the dry ingredients, then add one-third of the butter, and continue alternating the remaining flour and butter. Add the chocolate and 3 ounces of chopped peppermint bark and mix to combine.
Spread the batter evenly in the pan. Sprinkle remaining 2 ounces peppermint bark over top. Bake for 40-45 minutes if using a 9-inch pan, 55-65 minutes if using an 8-inch pan. A cake tester or wooden skewer poked into the center should come out with just a few moist crumbs sticking to it. If the pick comes out wet, test a second time, because you may have hit a piece of chocolate chip; then bake for a few more minutes longer if necessary.
Cool in the pan until the brownie is at room temperature. Run a knife around the edges, and invert the brownie onto a cutting board. Cut into 12 rectangles or 16 squares. Dust the tops with powdered sugar just before serving.
The brownies can be stored in an airtight container at room temperature for up to 3 days, but they also freeze really well!