Preheat the oven to 350 degrees. Butter a 9-inch square baking pan.
In a small bowl, whisk together the flour, cacao powder, and baking powder.
In a medium bowl, cream together the butter and sugar. Beat in the egg and vanilla extract and cream together until fluffy. Beat in the melted chocolate, and then the applesauce. On low speed, beat in the dry ingredients. Spread the mixture in the pan and bake until just set, around 25 minutes.