Bring a large pot of salted water to a boil. Add green beans and cook for 2 minutes, then add asparagus; continue cooking until vegetables are just tender, about 3 minutes more. Make sure they don’t overcook. You want them to have a nice ‘bite’.
Using a sieve, remove the vegetables from water (reserve water); run under cold water to stop cooking. Cut beans in half lengthwise. Cut asparagus in half crosswise, then quarter lengthwise.
In a large pot, melt the butter over medium heat; add pine nuts, shallot and a pinch of salt. Cook, stirring frequently, for about 1 minute, then add beans and asparagus, stir to combine and remove from heat. In a small bowl, whisk together the grated cheese, egg, egg yolks, parsley and a generous grinding of pepper.
Return vegetable cooking water to a boil. Add pasta and cook according to the packaging.
Drain the pasta and transfer to a large serving bowl; immediately add the vegetable mixture and egg mixture and toss well to coat strands. Serve immediately, passing extra cheese and pepper at the table.
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