Dice the onion, bell pepper, and celery into 1/4-inch pieces. Mince the garlic, slice the sausage into thin half-moons, and combine the Cajun seasoning, smoked paprika, thyme, and black pepper in a small bowl.
Boil the spaghetti in salted water for 1 to 2 minutes less than package directions. Reserve 1 cup of pasta water, then drain the pasta without rinsing.
Heat olive oil in a Dutch oven over medium-high heat. Brown the andouille sausage for 3 to 4 minutes, then transfer to a plate, leaving the drippings in the pot.
Add the ground beef to the same pot and break it into crumbles. Cook for 6 to 8 minutes until browned, then spoon out excess grease, leaving about 2 tablespoons behind.
Add the onion, bell pepper, and celery to the beef and cook for about 5 minutes until softened. Stir in the garlic for 30 seconds, then add the Cajun seasoning mixture and stir for 1 minute.
Push the meat and vegetables to the sides, add butter to the center, then sprinkle in the flour and stir everything for 2 minutes. Add tomato paste and stir for 1 minute, then slowly pour in the beef broth while scraping up browned bits. Stir in Worcestershire sauce, return the sausage, and simmer for 8 to 10 minutes until thickened.
Add the drained spaghetti to the pot and toss with tongs until coated. Pour in about 1/2 cup of reserved pasta water and toss for 2 minutes, adding more water if needed. Taste and season with salt and hot sauce.
Remove from heat and let rest for 2 minutes. Scatter green onions and parsley over the top and serve hot with extra hot sauce on the side.