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 with a sheet of parchment paper and butter the parchment.
In a medium bowl, whisk together 3/4 cups flour, 1/2 teaspoon salt, and 1 tablespoon dark unsweetened cocoa powder; set aside.
Set up a large double boiler (large heatproof bowl set over, but not touching, simmering water). Add 5 ounces coarsely chopped dark chocolate and 1/2 cup (1 stick) butter (cut into 1-inch cubes) to the bowl. Stir occasionally until the chocolate and butter are completely melted and smooth.
Turn off the heat under the double boiler but keep the bowl over the water. Add 3/4 cups sugar and 1/4 cup firmly packed light brown sugar to the melted chocolate mixture and whisk until fully combined. Remove the bowl from the pan and let the mixture cool to room temperature.
Add 3 large eggs (at room temperature) to the cooled chocolate mixture and whisk until just combined. Add 1 teaspoons vanilla extract and stir until combined. Do not overbeat.
Sprinkle the reserved flour/cocoa/salt mixture over the chocolate mixture. Using a spatula, fold the dry ingredients into the wet until only a trace of dry streaks remain.
Pour the batter into the prepared pan, smooth the top with an offset spatula, and bake for 12 to 15 minutes, rotating the pan halfway through baking. The brownies should be just underdone: a toothpick inserted at an angle should come out with a few loose crumbs. Remove from the oven and let the brownie layer cool completely in the pan.
To make the creme de menthe buttercream, in a medium heavy-bottomed saucepan whisk together 3/4 cups sugar and 2 tablespoons flour. Add 3/4 cup milk and 2 tablespoons heavy cream.
Cook the milk mixture over medium heat, whisking occasionally, until it comes to a boil and thickens, about 5 to 7 minutes. Remove from heat.
Transfer the hot mixture to the bowl of an electric mixer fitted with the paddle attachment. Beat on high speed until the mixture cools to room temperature (cooling fully is important so the butter does not melt).
Reduce the mixer speed to low and add 1 1/2 sticks (3/4 cup) butter, softened but still cool and cut into small cubes. Mix until the butter is thoroughly incorporated. Increase the speed to medium-high and beat until the filling is light and fluffy.
Add 3 tablespoons creme de menthe and 1 teaspoon peppermint extract and mix until evenly combined. If the filling is too soft, chill briefly in the refrigerator and then re-whip. If it is too firm, place the bowl over a pot of simmering water to warm slightly and then re-mix.
Spread the creme de menthe filling evenly over the completely cooled brownie layer. Place the pan in the refrigerator for a minimum of 45 minutes.
To make the chocolate glaze, set a large non-reactive metal bowl over a saucepan of simmering water. Combine 6 ounces coarsely chopped dark chocolate, 1 teaspoon light corn syrup, and 1/2 cup (1 stick) unsalted butter (softened and cut into cubes) in the bowl. Stir until the mixture is completely smooth.
Remove the bowl from the heat and stir vigorously for 1 minute to release excess heat. Pour the 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 hardened.
Remove the pan from the refrigerator and let it sit about 15 minutes to allow the glaze to soften slightly. Use a warm knife (dip in hot water, wipe dry) to cut the bars into squares. Store the bars tightly covered in the refrigerator for up to 4 days.