Combine the carrots, apples, raisins, and bread crumbs in a bowl. Sift together the flour, baking powder, cinnamon, baking soda, salt, nutmeg, and allspice in a separate small bowl. Cream together the butter, sugar, and eggs in a separate large bowl; stir in the flour mixture. Add the carrot mixture and mix well. Transfer mixture to a lightly-greased, 2-quart mold.
Place a steamer rack in the bottom of a large, deep pot. Set the filled mold on top of the steamer rack. Pour enough water into the bottom of the pot to cover the bottom 2/3 of the mold. Bring to a boil over medium heat. Reduce heat to low; cover. Simmer for 2 hours 45 minutes, adding water occasionally to maintain the water level. Remove from water and allow to cool 10 minutes before turning out of mold onto a plate.
