Preheat the oven to 325°F. Butter the sides and bottom of a 9-by-13-inch glass or light-colored metal pan. Line the bottom of the pan with a sheet of parchment paper, then butter the parchment.

In a medium bowl, whisk together the 3/4 cup flour, 1/2 teaspoon salt, and 1 tablespoon dark unsweetened cocoa powder until evenly combined. Set aside.

Set up a double boiler (a heatproof bowl set over, but not touching, simmering water). Place the 5 ounces coarsely chopped dark chocolate and 1/2 cup (1 stick) butter (cut into 1-inch cubes) in the bowl. Stir occasionally until fully melted and smooth.

With the double boiler still assembled but the heat turned off, add the 3/4 cup granulated sugar and 1/4 cup firmly packed light brown sugar to the melted chocolate mixture. Whisk until the sugars are completely incorporated. Remove the bowl from the hot water and let the mixture cool to room temperature.

When the chocolate mixture is at room temperature, whisk in the 3 large eggs one at a time just until combined, then stir in the 1 teaspoons vanilla extract. Do not overbeat.

Sprinkle the reserved flour/cocoa/salt mixture over the chocolate mixture. Using a spatula (not a whisk), gently fold the dry ingredients into the wet just until only a trace of the dry mixture is visible.

Pour the batter into the prepared pan and smooth the top with an offset spatula. Bake for 12 to 15 minutes, rotating the pan halfway through baking. The brownies should be slightly underbaked — about 1 minute shy of fully set. A toothpick inserted at an angle should come out with a few loose crumbs.

Remove the pan from the oven and let the brownie layer cool completely in the pan on a wire rack while you make the filling.

For the buttercream filling, combine the remaining 3/4 cup sugar and 2 tablespoons flour in a medium heavy-bottomed saucepan and whisk to blend. Add 3/4 cup milk and 2 tablespoons heavy cream and cook over medium heat, whisking occasionally, until the mixture comes to a boil and thickens, about 5 to 7 minutes.

Transfer the thickened milk mixture to the bowl of an electric mixer fitted with the paddle attachment. Beat on high speed until the mixture has cooled to near room temperature.

Reduce the mixer speed to low and add the 1 1/2 sticks (3/4 cup) softened but still cool butter, a few cubes at a time if needed, until incorporated. Increase the speed to medium-high and beat until the filling is light and fluffy.

Add the 3 tablespoons creme de menthe and 1 teaspoon peppermint extract to the buttercream and mix until evenly combined. If the filling is too soft to spread, chill briefly in the refrigerator and then beat again until spreadable. If it is too firm, warm the bowl gently by placing it over a pot of simmering water and re-whip to a spreadable consistency.

Spread the buttercream filling evenly over the cooled brownie layer. Refrigerate the pan for a minimum of 45 minutes to firm the filling while you make the glaze.

For the chocolate glaze, place 6 ounces coarsely chopped dark chocolate, 1 teaspoon light corn syrup, and 1/2 cup (1 stick) unsalted butter (softened, cut into cubes) in a large non-reactive metal bowl set over a saucepan of simmering water. Stir until the mixture is completely smooth and glossy.

Remove the glaze bowl from the heat and stir vigorously for 1 minute to release excess heat and ensure a smooth texture.

Pour the warm glaze over the chilled creme de menthe layer and use an offset spatula to spread it into an even layer. Return the pan to the refrigerator and chill for about 1 hour, or until the glaze has fully hardened.

Remove the pan from the refrigerator and let sit about 15 minutes at room temperature to slightly soften the glaze. Use a knife warmed under hot water and wiped dry between cuts to cut the bars into squares. Serve immediately or store.

Store any leftovers tightly covered and refrigerated for up to 4 days.