Preheat your oven to 350°F (175°C). Line a 9x13 inch baking pan with parchment paper, leaving an overhang on the sides to easily lift the brownies out later. Lightly grease the parchment paper.

In a large microwave-safe bowl or a saucepan over low heat, melt the butter. Remove from heat and stir in the granulated sugar until well combined.

Whisk in the eggs one at a time until fully incorporated. Stir in the vanilla extract.

In a separate bowl, whisk together the all-purpose flour, cocoa powder, and salt. Gradually add the dry ingredients to the wet ingredients, mixing with a rubber spatula until just combined. Be careful not to overmix.

Pour the brownie batter into the prepared baking pan and spread evenly. Bake for 30-35 minutes, or until a toothpick inserted into the center comes out with moist crumbs, not wet batter. Do not overbake. Let the brownies cool completely in the pan on a wire rack.

In a small bowl, combine the cooled espresso (or coffee), coffee liqueur (or dark rum), and optional granulated sugar. Stir until the sugar dissolves.

Once the brownies are completely cool, use a pastry brush to evenly brush the espresso mixture over the entire surface of the brownies. Allow the liquid to soak in for a few minutes.

In a large mixing bowl with an electric mixer, whip the very cold heavy cream and sifted powdered sugar on medium-high speed until stiff peaks form.

In a separate bowl, gently whisk the softened mascarpone cheese and vanilla extract until smooth. Be careful not to overmix mascarpone, as it can curdle.

Gently fold the whipped cream into the mascarpone mixture until just combined and smooth. Do not overmix.

Spread the mascarpone frosting evenly over the espresso-soaked brownies. Smooth the top with an offset spatula.

Cover the pan with plastic wrap and refrigerate for at least 2 hours, or preferably overnight, to allow the frosting to set and the flavors to meld.

Before serving, use the parchment paper overhang to lift the brownies out of the pan and place them on a cutting board. Dust generously with unsweetened cocoa powder using a fine-mesh sieve. Cut into 16 squares.