Preheat oven to 350 degrees F (175 degrees C). Grease an 8-inch baking pan.
Puree black beans, eggs, 1/2 cup agave nectar, 6 tablespoons cocoa powder, butter, 1 tablespoon coffee, vanilla extract, and salt in a food processor until smooth. Add hazelnuts and baking powder; pulse about 8 times. Pour cake batter into the baking pan.
Bake in the preheated oven until cake appears dry, with cracks forming in the middle, about 35 minutes.
Blend avocado, 1/4 cup agave nectar, 2 tablespoons cocoa powder, 2 1/2 teaspoons coffee, and hazelnut liqueur in the food processor until smooth. Spread icing over cake; cover cake with plastic wrap and refrigerate until completely cooled, at least 2 hours.