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1. Make the Brownies: Preheat oven to 350 degrees F. Butter an 8-inch-square baking dish and line with parchment paper, letting the excess hang over the edges. Butter the parchment paper; set aside.
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2. In a small bowl, whisk together the flour, baking powder, and salt; set aside.
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3. Bring about an inch of water to a simmer in a medium saucepan. Place a medium, heatproof bowl on top of the saucepan (do not let it touch the water) and add the butter and unsweetened chocolate. Stir occasionally, until the chocolate and butter are completely melted and smooth. Remove the bowl from the saucepan and whisk in the granulated and brown sugars, along with the instant espresso powder. Whisk in the egg, the yolk, and the vanilla extract. Add the flour mixture all at once, and fold it into the chocolate mixture with a rubber spatula until combined.
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4. Pour the batter into the prepared pan and bake for 25 to 30 minutes, or until a thin knife inserted in the center comes out clean. Allow to cool completely; meanwhile, make the frosting.
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5. Make the Frosting: Mix the instant espresso powder into the vanilla until dissolved; set aside. Using the whisk attachment of a stand mixer, whip the butter on medium-high speed for 5 minutes, stopping once to scrape the sides of the bowl. Reduce the mixer speed to low and add the powdered sugar a little at a time, waiting until it is mostly incorporated before adding more. Once all of the powdered sugar has been added, scrape the sides of the bowl and increase the speed to medium-high and whip until fluffy, about a minute or two. Add the espresso and vanilla mixture and continue to mix at medium-high until it is completely incorporated. Scrap the sides of the bowl, add the heavy cream, and again whip briefly at medium-high speed until thoroughly incorporated.
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8. Spread the frosting onto the cooled brownies, and cut into squares. Store brownies in an airtight container, or tightly wrapped, at room temperature for up to 5 days.
