How to Make It
Preheat the oven to 400°. Prick the potatoes once or twice with a small knife and bake for about 45 minutes, or until tender in the center when pierced. Let cool slightly.
Peel the potatoes and pass them through a potato ricer or food mil into a large bowl. Let cool completely. Stir in the salt. Gradually stir in about half of the flour until the mixture becomes stiff. Transfer the mixture to a lightly floured surface and knead in as much of the remaining flour as necessary until the dough is no longer sticky.
Line 2 baking sheets with wax paper and dust with flour. Divide the dough into 8 equal pieces. Set one piece on a floured surface; keep the remaining dough covered with an inverted bowl. using your hands, roll out the dough to a 1/2-inch-thick rope and then cut the rope into 3/4-inch lengths.
Dip a fork in flour. Roll each piece of dough down the back of the tines; press gently with your thumb so the tines leave ridges. Alternatively, flatten the gnocchi by pressing each one gently with a fork. Set the gnocchi on the prepared baking sheets in a single layer without touching. Repeat with the remaining pieces of dough.
Heat the olive oil in a large nonreactive skillet. Add the onion and cook over moderately high heat, stirring, until tender, about 5 minutes. Add the tomatoes and season with salt and pepper. Bring to a simmer, then reduce the heat to low and cook for 20 minutes, stirring occasionally.
Bring a large pot of water to a boil. Stir in 2 tablespoons of salt, then add half the gnochhi, a few at a time, and simmer until they float to the surface. Using a slotted spoon, transfer them to a rack to drain thoroughly; don't pile them or they'll stick together. Repeat with the remaining gnocchi.
Stir the prosciutto and basil into the tomato sauce. Spoon the gnocchi onto 4 large plates, top with the tomato sauce and serve.