Chicken and Mushroom Tagliatelle

Name: Chicken and Mushroom Tagliatelle
Ingredients: 12 ounces fresh or dried tagliatelle; Coarse salt and freshly ground pepper; ¾ pound boneless, skinless chicken thighs; 2 tablespoons unsalted butter; 2 shallots, halved, peeled, and thinly sliced (about ½ cup); 8 ounces cremini mushrooms, halved (or quartered if large); ½ cup dry white wine; ¾ cup heavy cream; 1 ounce Parmesan, grated (about ⅓ cup), plus more for serving; 1 stalk celery with leaves, cut into very thin slices (about ⅔ cup); ⅓ cup coarsely chopped fresh flat-leaf parsley leaves