Balsamic Mushroom and Sausage Pasta

Name: Balsamic Mushroom and Sausage Pasta
Ingredients: 12 oz. medium tube pasta (such as mezzi rigatoni or pipe rigate); 1½ tsp. Diamond Crystal or ¾ tsp. Morton kosher salt, divided, plus more; 4 Tbsp. extra-virgin olive oil, divided; 4 Tbsp. unsalted butter, divided; 1 lb. crimini or button mushrooms, cut into ¼ pieces; 8 oz. hot Italian sausage; 1 large red onion, finely chopped; 4 garlic cloves, coarsely chopped; 2 Tbsp. tomato paste; ¼ tsp. crushed red pepper flakes; Freshly ground pepper; ⅓ cup balsamic vinegar; Finely grated Parmesan (for serving)