Preheat an oven to 350 degrees F (175 degrees C). Grease three baking sheets.
Process the cereal in a food processor or blender until a fine crumb is achieved. Whisk the cereal crumbs together with the bread flour, baking powder, and baking soda in a mixing bowl; set aside.
Beat the butter, shortening, brown sugar, and white sugar with an electric mixer in a large bowl until smooth. Add 1 egg and allow to blend into the butter mixture before adding the other with the vanilla. Mix in the flour mixture until just incorporated. Drop by heaping spoonfuls onto the prepared baking sheets.
Bake in the preheated oven until the edges are golden, about 12 minutes. Allow the cookies to cool on the baking sheet for 1 minute before removing to a wire rack to cool completely.