Place the ham and salt in a large pot and cover with water. Bring to boil. Reduce the heat and simmer until the ham is tender, about 2 hours.

Place the apples in large bowl, cover with water, and soak for 2 to 3 hours. When the ham is tender, add the apples and their soaking liquid to the pot. Add the brown sugar and onions, and bring to a boil. Reduce the heat and simmer, covered, for 1 1/2 hours.

Into a large bowl, sift together the flour, baking powder, salt, and pepper. Add the egg and melted butter, and enough milk to make a moist, fairly stiff batter.

Uncover the ham and apples, and bring the cooking liquid to a boil. Drop the batter by tablespoonfuls into the boiling liquid. Cover tightly and simmer for 12 to 15 minutes. Season to taste with salt and freshly ground black pepper.

To serve, transfer the ham to a large platter and let sit for 5 minutes. Carve thinly and spoon the dumplings around the meat. Serve.