Place all gnudi ingredients in a bowl and combine with a spoon until a rough dough forms. Turn out on to a lightly floured work surface and knead with your hands until well combined and slightly less tacky. Return to bowl and cover with plastic film. Refrigerate for at least 30 minutes. This will tighten up the dough and allow you to work with it more easily.
When ready to cook, remove from the refrigerator and shape however you wish. While the quick approach is to shape into balls, the traditional way is to make cylinders by rolling out the gnudi into a rope shape and cutting segments out of it. The key to a good shape is size. If they are too small, you will not have much of a nice melted center. If they are too large, it may not be fully cooked in the center, resulting in a cold center with an uncooked flour taste. Pieces about half the size of a golf ball work best.
To cook the gnudi, place a pan over medium heat with enough olive oil to lightly coat the entire surface. Sauté the gnudi gently, rolling them over a bit every once in a while so that you get nice golden spots along the surface. Cook until completely tender and melty. (Yes, that’s a word.)
In a separate pan, also with olive oil on medium heat, toss in some hand-shredded prosciutto or other cured ham (Serrano, speck, guanciale, pancetta, etc.—smoked bacon also works well). Gently sweat the shredded ham until it renders out some fat and flavor into the pan. Add 1 to 3 knobs of butter, depending on portion size, and some water. Cook everything together, swirling the pan until a nice butter sauce forms in the pan. Be careful not to cook it at too high of a heat or the butter will break and you will get an oily sauce. Taste the sauce and season to taste with salt and pepper.
Place the golden gnudi in a shallow bowl and pour over the buttered ham sauce. Toss the arugula with a little olive oil and lemon juice and garnish. Enjoy.