Bring a large pot of salted water to a boil over high heat. Add the pasta and cook 2 minutes less than the package directions, about 8 minutes. Drain well, reserving 1 1/2 cups of the pasta cooking water.
Meanwhile, heat a large straight-sided skillet over medium heat. Add the olive oil and pancetta. Cook, stirring often with a wooden spoon, until the pancetta is crisp, about 10 minutes. Add the pepper and toast it, stirring often, for about a minute, or until fragrant. Add 1/2 cup of the reserved pasta water and scrape up any bits that are sticking to the bottom of the skillet.
Add the pasta and sprinkle with the parmigiano-reggiano. Add another 1/2 cup of the reserved pasta water and stir to coat all of the pasta in the cheese. Add the pecorino and butter and stir to combine, creating a light, creamy cheese sauce. Add the arugula and cook until wilted. Add additional reserved pasta water as needed to maintain the light sauce consistency. Serve with more grated parmigiano-reggiano, if desired.
Reprinted From Giada's Italy: My Recipes For La Dolce Vita By Giada De Laurentiis. Copyright © 2018 By GDL Foods Inc. Published by Clarkson Potter/Publishers, an imprint of Penguin Random House LLC.