Wash the potatoes and place in a small saucepan. Cover with cold water and 1 teaspoon of the salt and bring to a boil over high heat. Reduce the heat to medium-low and simmer for 30 to 35 minutes until the potatoes are tender. Drain.
Peel potatoes while they are still warm. (Not too hot, or you might burn your hands.) Place the peeled potatoes in a ricer over a small bowl and push through. Alternatively, using the back of a spoon or a rubber spatula, mash the potatoes through the holes in a colander into a bowl set below it. You can mash the potatoes by hand, but the dumplings won't be as fluffy.
Add the ham, Parmesan, beaten eggs, butter, nutmeg, and the remaining 1 teaspoon of salt to the mashed potatoes. Mix well to combine.
Line a small baking sheet with paper towels, and place the flour in a small mixing bowl.
Roll the potato mixture into little balls, about 2 rounded teaspoonfuls each. Toss each dumpling in the flour, shake off the excess flour and place on the prepared baking sheet. Set aside.
In a 6-quart soup pot or stockpot, combine the beef broth, chicken broth, water, onion, carrots and parsley. Bring to a boil and boil for 10 minutes. Turn off the heat and cover to keep warm.
Heat 2 tablespoons of the oil in a large nonstick skillet over medium to medium-high heat. Add 1/3 of the potato dumplings to the skillet, leaving enough room to turn them. Brown the dumplings on all sides, 6 to 8 minutes, until golden, and transfer to a paper towel-lined plate or baking sheet. Continue to brown the remaining dumplings, using 2 tablespoons of the oil for each batch.
Heat the beef-chicken stock over medium-high until simmering, over medium-high heat. Using a ladle, carefully place the dumplings into the simmering stock. Simmer dumplings for 3 to 5 minutes. Serve immediately since the dumplings are fragile.
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