Spicy Rigatoni With Pork Sugo

Name: Spicy Rigatoni With Pork Sugo
Ingredients: 1 large dried shiitake mushroom, cut into 1 pieces; 2 medium onions, cut into 1 pieces; 6 garlic cloves; 1 medium carrot, peeled, cut into 1 pieces; 8 oz. bacon slices, finely chopped; ½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt, plus more; ½ cup chicken livers (3–4), rinsed; 1 Tbsp. Thai chile flakes or 2 Tbsp. crushed red pepper flakes; 2 Tbsp. smoked paprika; 1 Tbsp. freshly ground pepper; ½ cup dry white wine; 1½ lb. ground pork; 1 28-oz. can crushed tomatoes; 1 qt. low-sodium chicken broth; ¼ cup fish sauce (preferably Three Crabs); ¼ oz. Parmesan, finely grated, plus more for serving; 1 lb. rigatoni; 2 Tbsp. chilled unsalted butter