Turkey Etcetrazzini

Name: Turkey Etcetrazzini
Ingredients: 2 Tbsp. extra-virgin olive oil or vegetable oil; 2 Tbsp. unsalted butter; 1 medium onion, coarsely chopped; 1 lb. mushrooms (any kind), torn or chopped into bite-size pieces; Freshly ground black pepper; 1½ tsp. Diamond Crystal or 1 tsp. Morton kosher salt, plus more; 6 sprigs thyme, leaves picked; 1 bay leaf; ½ cup dry white wine; 1½ cups heavy cream; 3 cups shredded leftover roast turkey meat; 1 lb. short pasta (such as farfalle, fusilli, or orecchiette); 2 cups leftover Thanksgiving sides (such as roasted brussels sprouts, sautéed green beans, or cooked greens; optional); 2 oz. finely grated Parmesan (about 1 cup), divided