Heat oven to 350 degrees. Butter an 8-inch square pan; set aside. In a medium saucepan over low heat, melt butter. Remove pan from heat, and stir in cocoa. Let cool slightly.
Meanwhile, in a small bowl, whisk together flour, baking powder, and salt.
Stir in sugar and sweet-potato puree, then egg. In a small bowl, stir together vanilla and coffee until coffee is dissolved; add to cocoa mixture.
Add flour mixture to cocoa mixture and stir until no traces of flour remain. Spoon into prepared pan; smooth the top. Bake until surface of brownies looks barely dry and an inserted knife comes out with a few moist crumbs, about 20 minutes. Cool to room temperature before serving.