Mix the flour with the eggs, egg yolk, and 1 teaspoon of oil until you obtain a dough. Wrap it in plastic wrap and let it rest for 1 hour. Grate the pecorino and set aside.
Chop the walnuts and toast them in a small pan. Peel the fava beans and cook them for 5 to 6 minutes in boiling salted water with the peeled garlic cloves. Drain, reserve 2 tablespoons of fava beans, and blend the rest with the garlic, yogurt, toasted walnuts (keep some for decoration), a few basil leaves, and 2 tablespoons of oil.
Flour your work surface and roll out the dough thinly with a rolling pin. Cut the rolled-out dough into wide strips. Bring a large pot of salted water to a boil and add the basil sprigs tied with kitchen string; remove the basil and cook the pappardelle in the boiling water until they float to the surface.
Dilute the fava bean cream with a bit of pasta cooking water. Drain the pappardelle and place them in a large dish, toss them with the sauce, the remaining fava beans, and three-quarters of the pecorino, mixing gently. Finish with a few shavings of pecorino, the remaining walnuts, a drizzle of oil, some ground pepper, and a few basil leaves.