Preheat oven to 350 degrees F (175 degrees C). Grease and flour an 8-inch square pan.
Melt 1/4 cup butter in a large saucepan over low heat. Remove from heat and stir in white sugar, eggs, applesauce, and 1 teaspoon vanilla extract. Beat in 1/3 cup cocoa powder, flour, salt, and baking powder. Spread into prepared pan.
Bake in preheated oven until brownies are just set in the center, 20 to 25 minutes.
Combine 3 tablespoons butter, 2 tablespoons cocoa powder, honey, 1 teaspoon vanilla extract, and confectioner's sugar in a bowl; beat until smooth. Frost brownies while they are still warm.
