Preheat your oven to 350°F (175°C). Lightly grease a 9x13 inch baking pan with butter or cooking spray, then line it with parchment paper, leaving an overhang on the sides for easy lifting. This helps prevent sticking and makes removal simple.

In a large microwave-safe bowl or small saucepan, melt the butter. Stir in the granulated sugar until well combined. The mixture should be warm, not boiling hot.

Allow the butter-sugar mixture to cool for a few minutes. Whisk in the eggs one at a time until fully incorporated and the mixture is smooth and glossy. Stir in the vanilla extract.

In a separate medium bowl, whisk together the all-purpose flour, unsweetened cocoa powder, baking powder, and salt. Ensure there are no lumps of cocoa powder.

Gradually add the dry ingredients to the wet ingredients, mixing with a rubber spatula until just combined. Be careful not to overmix; a few streaks of flour are okay. Overmixing can lead to tough brownies.

Pour the brownie batter into the prepared 9x13 inch baking pan and spread evenly. Bake for 25-28 minutes, or until a toothpick inserted into the center comes out with moist crumbs, but not wet batter. Do not overbake for fudgy brownies.

Remove the brownies from the oven and let them cool in the pan on a wire rack for at least 15-20 minutes before proceeding to the coffee soak. They should still be warm but not piping hot.

While the brownies are cooling, prepare the coffee soak. In a small bowl, combine the strong brewed coffee (or espresso), optional granulated sugar, and optional coffee liqueur. Stir until the sugar is dissolved.

Using a toothpick or skewer, poke holes evenly across the top of the warm brownies (about 1 inch apart). Slowly and evenly pour the coffee mixture over the poked brownies, allowing it to soak in. Let cool completely to room temperature.

In a large mixing bowl, beat the cold mascarpone cheese with an electric mixer on medium speed for about 1 minute until slightly softened and smooth. Be careful not to overmix, as mascarpone can curdle.

Add the cold heavy whipping cream, sifted powdered sugar, and vanilla extract to the mascarpone. Start mixing on low speed, then gradually increase to high speed. Whip until the mixture is thick, fluffy, and holds stiff peaks.

Once the coffee-soaked brownies are completely cool, spread the mascarpone frosting evenly over the top using an offset spatula or the back of a spoon.

Cover the pan with plastic wrap and refrigerate for at least 2-3 hours, or preferably overnight, to allow the frosting to set and the flavors to meld. Just before serving, dust the top generously with unsweetened cocoa powder using a fine-mesh sieve.

Lift the brownies out of the pan using the parchment paper overhang. Cut into 16 squares and serve chilled.