Preheat an oven to 350 degrees F (175 degrees C). Prepare a 2-quart baking dish with cooking spray.
Place the dried mushrooms in a bowl.
Pour 1/4 cup of warm water over the mushrooms; allow to soak until the mushrooms have rehydrated, about 30 minutes.
Drain, reserving the water, and set both the mushrooms and water aside.
Combine the ground pork, carrots, onion, garlic, 2 tablespoons soy sauce, the drained mushrooms, and the egg white in a large bowl; mix with hands until evenly combined. Form the mixture into 16 meatballs; arrange into the prepared baking dish.
Stir the reserved water from the mushrooms, the applesauce, 1/4 cup water, maple syrup, 2 tablespoons of soy sauce, and brown sugar together in a bowl; pour over the meatballs. Cover the dish with aluminum foil.
Bake in the preheated oven for 45 minutes.
Remove the foil from the dish and continue baking until the meatballs are cooked through and no longer pink in the center, about 20 minutes more.
