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Preheat oven to 350F. Line an 8-inch square pan with parchment paper.
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Melt ½ cup butter. Remove from heat, and stir in sugar, eggs, and vanilla. Beat in cocoa powder, flour, salt, and baking powder. Spread batter into prepared pan.
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Bake in preheated oven for 25 to 30 minutes, or until a toothpick stuck in the middle comes out clean.
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To make the frosting:
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Beat the butter until fluffy.
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Dissolve the instant coffee in the hot water. Add to the butter, along with the vanilla and cocoa powder. Beat until combined. Start adding powdered sugar, ¼ cup at a time, beating after each addition until it reaches the desired consistency. (You can use more powdered sugar if you prefer a stiffer buttercream.)