Preheat the oven to 325 degrees. In a large bowl, beat together the applesauce, honey, egg, egg white, and vanilla.
In a small bowl, combine the carob, flour, and baking powder. Add to the applesauce mixture. Stir in the nuts. Pour the mixture into an oiled 8- by 8-inch baking pan. Bake for 20 to 25 minutes. Remove from the oven and cool completely. Cut into squares.