Place the ham shank in a large Dutch oven or pot, cover with water, and bring to a boil. Reduce the heat and simmer for 1 ½ hours (add more water if necessary to always keep the ham covered)
Remove the ham and add the cubed potatoes, carrots, and celery to the broth. Season with salt and pepper
Cube ham and set it aside. Discard the bone
While the veggies are cooking, make the dough by mixing the flour, eggs, and milk until a soft dough forms
Roll out dough on the counter to ¼ inch thick and cut into small squares
With the soup boiling, add the squares and cook them until fully cooked. Return the diced ham to the pot, correct the seasoning, and add salt and pepper if needed. Cook until heated through. Serve with chopped parsley on top