For the fresh pasta: On a clean work surface, make a mound with the flour, then create a well. Pour the eggs into the well and use a fork to gradually flick the flour into the eggs, whisking in thoroughly as you go. Once all of the flour and eggs have been incorporated into a very shaggy dough, form it into a ball with your hands. Knead the dough, dusting with more flour if the dough is too sticky, or moistening your hands with water if the dough is too dry, until it has the texture of smooth leather, about 5 minutes. Wrap it in plastic wrap and leave at room temperature for 25 minutes to rest.
Use your pasta machine to roll out the dough and cut fresh linguine, following the pasta machine's instructions. Dust with flour and cover the freshly-cut noodles with a damp towel while you work, and until ready to cook.
Bring a large pot of water to a rolling boil, and season with a handful of salt.
For the sauce: Heat a large saute pan over medium-high heat. Add the butter, and once it melts, add the asparagus, peas, string beans, garlic scapes, onions and a pinch of salt. Cook just until the onions soften and the scapes are fragrant, about 5 minutes. Add the cream and chicken stock and cook to reduce, about 3 minutes.
Add the fresh pasta to the pot of boiling water, and cook until the pasta is just al dente, 2 to 3 minutes. Remove the pasta and transfer to the pan with the primavera sauce, then transfer everything to a serving bowl. Toss everything to coat, and taste for seasoning. Garnish with fresh Parmesan and basil and serve.