Meanwhile melt butter in a large frying pan over low heat. With tongs, add pasta directly to the pan. Add Parmigiano-Reggiano, 1/2 cup reserved cooking water, and 1/2 tablespoon salt to pot. Using tongs, toss and stir pasta vigorously to thoroughly combine, about 1 minute. Add additional pasta water as necessary; the sauce should be creamy and emulsified and coat each strand of the pasta. Remove pot from heat, cover, and let pasta sit for 1 minute. Season with salt to taste.