Brown the Andouille Sausage: In a large heavy-bottomed pot or Dutch oven, heat 1 tablespoon of olive oil over medium-high heat until shimmering. Add the sliced andouille sausage and cook, stirring often, until the sausage browns and renders its fat, about 8-12 minutes. Transfer the sausage to a clean plate.
Add the Pork and Season: Return the pot to medium-high heat (add an additional tablespoon of olive oil if needed). Add the diced pork shoulder (or chicken thigh) and season with approximately ½-1 tablespoon of Cajun seasoning, salt, and black pepper (about ½ teaspoon of each).
Brown the Pork: Brown the pork for about 15 minutes, stirring only occasionally to make sure some searing takes place. Transfer the pork to a clean plate and set aside.Note: This will create a nice brown layer in the bottom of your pot- that's a good thing! Reduce the heat to medium if it starts to burn. Optional: Drain all but one tablespoon of the rendered fat/oil. Once the meat is removed to a clean plate, you'll have at least a few tablespoons of bright orange-colored grease. I usually drain most of it, leaving only about one tablespoon left in the pot.
Cook the Trinity: Lower heat to medium-low and add 2 tablespoons of butter to the pot. Stir in celery, onion, and bell pepper. Cook slowly until softened and golden, about 10-12 minutes, scraping the bottom of the pot as they cook.
Add the Aromatics: Add the minced garlic, thyme, remaining Cajun seasoning, and cayenne (if using), and cook for 1–2 minutes until fragrant.
Add the Meat Back to the Pot. Return the browned andouille sausage and pork to the pot. Mix well to combine.
Build the Sauce: Pour in the crushed tomatoes and chicken broth, and stir well, scraping up any browned bits still stuck on the bottom of the pot. Bring the mixture to a simmer.
Add the Pasta and Cook: Stir in the penne pasta, fresh parsley, and green onions. Reduce the heat to medium-low, and cover the pot. Cook for 10-12 minutes, stirring occasionally to prevent burning (reduce the heat to low if necessary). Continue to cook the pasta until tender and most of the liquid is absorbed.Tip: If needed, add a bit more broth or water to adjust the consistency. Finish with the Cream and Parm: Once the pasta is cooked, remove from heat and stir in the heavy cream (if using) and freshly grated parmesan cheese.
Season and Serve: Taste and adjust seasoning with salt, pepper, or extra Cajun spice. Serve hot, garnished with additional parmesan cheese, green onions, or parsley.