Step 1
Preheat oven to 350°. Bring 1½ cups water to a boil in a small saucepan. Place 1 large dried shiitake mushroom, cut into 1″ pieces, in a small bowl and pour water over. Let sit 30 minutes.
Step 2
Drain mushroom and transfer to a food processor. Add 2 medium onions, cut into 1″ pieces, 6 garlic cloves, and 1 medium carrot, peeled, cut into 1″ pieces, and pulse until almost a paste.
Step 3
Cook 8 oz. bacon slices, finely chopped, in a dry large Dutch oven or other heavy pot over medium, stirring occasionally, until starting to brown around the edges, about 5 minutes. Add vegetable paste and ½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt and stir to combine. Cook, scraping up browned bits and stirring occasionally, until paste is translucent, 6–8 minutes.
Step 4
Meanwhile, purée ½ cup chicken livers (3–4), rinsed, in food processor until smooth; set aside.
Step 5
Add 1 Tbsp. Thai chile flakes or 2 Tbsp. crushed red pepper flakes, 2 Tbsp. smoked paprika, and 1 Tbsp. freshly ground pepper to pot and cook, stirring constantly (you don’t want paprika to stick to bottom of pot and burn), until fragrant, 2–3 minutes. Add ½ cup dry white wine, bring to a simmer, and cook, stirring occasionally, until evaporated, about 2 minutes. Add 1½ lb. ground pork, one 28-oz. can crushed tomatoes, 1 qt. low-sodium chicken broth, and reserved liver and bring to a simmer, breaking pork into smaller pieces and mixing to combine.
Step 6
Carefully transfer pot to oven and place a baking sheet on rack below pot to catch any drips or spatters. Bake sugo, uncovered, scraping sides of pot with a wooden spoon or heatproof rubber spatula and incorporating browned bits back into sauce and breaking up any large pieces of pork every 30 minutes, until deepened in color and thickened, 2½–4 hours. Remove pot from oven and let sugo cool 20 minutes.
Step 7
Add ¼ cup fish sauce and ¼ oz. Parmesan, finely grated, to sugo and mix to incorporate, breaking up any remaining large pieces of pork. (You should have about 2 quarts sauce.) Transfer 1 quart sugo to an airtight container and let cool; reserve for another use.
Step 8
Cook 1 lb. rigatoni in a large pot of boiling salted water, stirring occasionally, until al dente. Drain, reserving 1 cup pasta cooking liquid.
Step 9
Add pasta to pot with remaining sugo and add 2 Tbsp. chilled unsalted butter and ½ cup pasta cooking liquid. Cook over medium-high heat, tossing pasta often and adding more pasta cooking liquid as needed, until butter is incorporated, liquid is almost completely evaporated, and pasta is al dente, about 3 minutes.
Step 10
Divide pasta among plates and top with more finely grated Parmesan.
Do ahead: Sugo can be made 3 days ahead. Let cool; cover and chill, or freeze up to 1 month.
