Bring a large pot of water with 2 tablespoons salt to a boil. Add the spaghetti and cook for 8 minutes, stirring occasionally. Reserve a cup of the pasta water, then add the snow peas, fresh peas, and asparagus to the spaghetti and cook for 2 minutes longer. Drain the pasta and vegetables together.
Meanwhile, heat the oil in a medium (10 to 11-inch) sauté pan over medium heat, add the pancetta, and cook for 7 to 9 minutes, stirring occasionally, until browned. Transfer the pancetta to a plate lined with paper towels and set aside.
While the pancetta cooks, fill a large bowl with the hottest tap water and set aside to heat the bowl. Just before you drain the pasta, pour the water out of the bowl. Put the cream, eggs, egg yolks, and 1/4 cup of the reserved pasta water into the bowl and whisk to combine. Immediately, add the hot pasta and vegetables and toss with tongs for a full minute or two, until the pasta absorbs the sauce. Add enough reserved hot pasta water to keep the sauce creamy. Add the 3/4 cup Parmesan, the scallions, chives, lemon zest and juice, 1 tablespoon salt, and 1 teaspoon pepper and toss well. Add the pancetta, sprinkle with salt, and serve hot sprinkled with extra chives and Parmesan.