In a small bowl, whisk together the whole egg, egg yolks, Parmigiano-Reggiano, and nutmeg. Remove the spaghetti from the heat and carefully drizzle in the egg mixture, tossing the noodles as you go to prevent the egg from scrambling, until fully incorporated. Add a splash of the reserved pasta cooking water and toss until the pasta looks silky. Season with salt and a generous amount of pepper, then stir in the butter, parsley, and lovage, if using. Transfer to a platter or serving plates and serve warm.