Preheat oven to 350 F. Grease a covered baking dish. I use a 2 quart or 2.25 quart dish.
In a large bowl, stir together bread crumbs, brown sugar, baking powder, baking soda, salt, cinnamon, allspice, and nutmeg.
Stir in egg.
Add raisins, potato, carrot, and apple. Stir to combine.
Stir in butter.
Pour carrot mixture into baking dish.
Cover, and bake for about 40-50 minutes.
To make the vanilla sauce, whisk together sugar, cornstarch, water, and salt in a saucepan.
Heat, stirring regularly, until thickened.
Remove from heat, and stir in butter and vanilla.
Serve pudding warm topped with vanilla sauce.
Notes
Bread crumbs:Coarse plain bread crumbs like panko bread crumbs work well.
Raisins: I prefer golden raisins in this recipe, but you can also use dark raisins if you prefer.
Potato: I measure the potato once it has been grated, not before.
Carrot: I measure the carrot once it has been grated.
Apple: I use a baking apple, like a Granny Smith, in this recipe. Also, I measure the apple once it has been diced.
Butter: I use salted butter in this recipe.
Covered dish: If you don't have a dish with a tight-fitting, oven-safe lid, you can wrap your baking dish tightly with aluminum foil. Just be careful when uncovering.