Preheat oven to 350°F (180°C). Grease and line an 8x8-inch pan with parchment paper, letting the excess extend over the sides of the pan. Set aside.
In a large bowl, whisk together the sugar, flour, cocoa powder, baking powder, and salt.
In a medium bowl, whisk together the butter, eggs, and vanilla until well combined. Add the butter mixture to the dry ingredients and mix until just combined. Pour the brownie batter into the prepared pan and spread it into an even layer.
Bake for 30 to 33 minutes. Test the brownies with a toothpick at the 30-minute mark. Insert it into the center of the pan; if it comes out with wet batter, the brownies are not done. If there are only a few moist crumbs, the brownies are done. Keep checking every 2 minutes until you have moist crumbs. Allow brownies to cool completely to room temperature in the pan before proceeding.
To Make the Peppermint Filling:
Place powdered sugar, butter, peppermint extract, and vanilla bean paste in a medium bowl. Add 3 tablespoons (45 grams) of boiling water, and stir until incorporated. (It should resemble a thick paste). Add additional boiling water 1 teaspoon (5 grams) at a time, if needed, to help get to correct consistency.
Immediately top the cooled brownie layer with the peppermint filling. Use a damp offset spatula to help smooth the peppermint filling into an even layer. Refrigerate until set, about 30 minutes.
To Make the Chocolate Topping:
In the top of a double boiler, combine chocolate and butter. Cook over simmering water, stirring frequently, until chocolate is melted and mixture is smooth.
Immediately top the chilled, set peppermint filling layer with the chocolate topping. Slam pan on a kitchen towel-lined counter to release any air bubbles. Refrigerate until topping is set, about 2 hours.
Using excess parchment as handles, remove from the pan and cut into bars. Serve chilled or at room temperature.
These brownies should be refrigerated but can be enjoyed chilled or at room temperature. Store leftovers in an airtight container in the refrigerator for up to 3 days.