Heat the oil in a heavy large frying pan over medium heat. Add the pancetta and garlic and sauté until it is brown and crisp, about 8 minutes. Cool slightly.
In a large bowl, whisk together the cream, cheese, yolks, basil, and parsley to blend.
Meanwhile, bring a large pot of salted water to a boil over high heat. Add the spaghetti and cook until it is just tender but still firm to the bite, stirring occasionally, about 10 minutes. Drain.
Add the chicken to the pan with the pancetta and stir to combine.
Next, add the spaghetti and the cream mixture and toss over medium-low heat until the chicken is heated through and the sauce coats the spaghetti thickly, about 4 minutes (do not boil or you might end up scrambling the eggs). Season the pasta, to taste, with salt and pepper if needed.
Transfer the pasta to a large wide serving bowl. Top with toasted walnuts and lemon zest.