Make gnocchi: Cut potatoes into chunks and pass through a food mill or potato ricer onto a lightly floured surface. Make a well in center of potatoes and add yolk to well. Sprinkle one-third of the flour over potatoes and combine potatoes, yolk and flour, mixing with your hands. Add another third of flour, kneading, then remaining flour, until a smooth, slightly sticky dough is formed. Divide dough in 4 pieces and roll each into a 15-inch-long rope, about 3/4-inch in diameter. Using a floured knife, cut each rope into 30 pieces. Dip tines of a dinner fork in flour and gently roll dough several times back and forth to form decorative ridges on one side.
Make tomato sauce: In a saucepan sweat onion and garlic in oil over medium heat until onions are golden and tender, 10 minutes. Add tomatoes and juice; bring to a boil and cook 3 minutes. Add basil and season with salt and pepper to taste.
Bring a large pot of salted water to a boil and drop in gnocchi. Cook until they rise to surface, about 2 minutes, and continue cooking 30 seconds more. Drain in a colander. In a large skillet, melt butter, add gnocchi, pour tomato sauce over, and toss to combine. Sprinkle with Parmesan and serve immediately.
Yield: 4 servings