Preheat the oven to 400 degrees . In a large bowl, combine the first 4 ingredients and the crushed red pepper, if using. Roll into 18 dumplings, each the size of a walnut.
In a medium pot, bring the stock and 3 cups water to a boil. Add the dumplings; simmer, covered, turning occasionally, until cooked through, about 15 minutes.
Meanwhile, set a wire rack over a rimmed baking sheet. Arrange the prosciutto slices in a single layer on the rack. Bake until crispy, 6 to 8 minutes. Transfer to a work surface and let cool. Crumble the prosciutto.
Using a slotted spoon, carefully transfer the dumplings to a shallow bowl; reserve the cooking liquid in the pot. In a large bowl, whisk 2 cups of the cooking liquid and the mashed potatoes; pour back into the pot. Bring the soup to a boil, stirring often.
Add the turkey and green beans to the soup and simmer until just heated though, 2 to 3 minutes. Divide the dumplings among bowls, ladle the soup over the dumplings and sprinkle with the crumbled prosciutto.