Place a large pot of water over high heat and bring up to a boil to cook the pasta. Once at a boil, add some salt and the pasta, and cook until al dente, according to package directions. Right before draining, remove and reserve 1 cup of the pasta-cooking water. Drain pasta thoroughly, return to the pot and reserve.
While the pasta is cooking, place a large skillet over medium-high heat with 1 turn of the pan of EVOO, about 1 tablespoon. Season the chicken with salt and ground black pepper, and cook in the pan until golden brown and cooked through, 3-4 minutes per side. Remove from the pan and set aside. When cool enough to handle, chop into bite-size pieces.
Add the dressing to the pan of cooked pasta along with the Parmigiano Reggiano and reserved cup of pasta water.
Toss with the lettuce and chopped chicken, and serve.