Preheat the oven to 350 degrees. Line an 8 or 9 inch square baking dish with a parchment sling and spray with a neutral cooking spray. Set aside.
In a medium sized pot, melt the butter on the stove top over medium heat. Remove from heat, then whisk in brown and white sugar until smooth and creamy, about 1-2 minutes.
Add the eggs and egg yolk, one at a time, whisking well after each addition. Stir in the vanilla.
Add the cocoa powder, flour, salt, and baking powder and stir until just combined.
Pour the mixture into the prepared pan and bake for 30-35 minutes or until center is just set. A toothpick inserted in the center should come back with a few moist crumbs attached.
In a measuring cup, mix together the espresso and dark rum.
Prick the warm brownies all over with a fork. Slowly pour the rum mixture over the brownies, allowing it to soak in. You don't want it to be puddled on top. You might have leftover coffee mixture, which is fine. Allow the brownies to cool.