Preheat oven to 350°F (180°C). Grease a 20x20cm (8x8 inch) baking dish.
In a small saucepan, mix ¾ cup dark or light brown sugar, ¼ cup unsweetened cocoa powder, a large pinch of salt, 1 ⅔ cups hot prepared coffee and 1 tablespoon vanilla extract. Place over medium heat and stir until almost boiling. Remove from the heat and set aside.
In a large bowl, mix together 1 cup all-purpose flour, ¾ cup sugar, ¼ cup unsweetened cocoa powder, 2 teaspoons baking powder, and ½ teaspoon salt.
Add ½ cup milk, ½ cup butter, melted, and ½ teaspoon vanilla extract to the dry ingredients and mix until just combined.
Spread the brownie batter in the prepared baking dish, smoothing to cover the edges.
Transfer the hot chocolate liquid mixture to a jar for easier pouring, and drizzle over the brownie batter, covering it completely. Do so slowly so you don't disturb the chocolate layer much.
Bake for about 40 minutes, until the liquid is absorbed and the surface is dark. When you jiggle the pan, you should notice the bottom is still liquid; this is correct.
Let cool slightly or serve warm with vanilla ice cream or whipped cream. Personally, I find it's better the next day, when the sauce has thickened. I rewarm it before serving.