Heat 1 tablespoon of olive oil over medium heat. Sprinkle 1 tablespoon of Cajun seasoning over diced chicken pieces and place half of the chicken in the pan (do not crowd). Brown chicken on all sides, remove from pan and repeat (adding last tablespoon of olive oil to pan) with the remaining chicken.
After removing all the chicken pieces, sauté sausage pieces 2-3 minutes or until lightly browned. Remove from pan.
Sauté onion, peppers and celery in the last tablespoon of olive oil for 3-4 minutes; add garlic and continue cooking for another minute, stirring constantly.
Add thyme, cayenne, black pepper, basil, 1 teaspoon Cajun seasoning, celery salt and bay leaf; stir until fragrant (about one minute).
Stir in Worcestershire sauce, Tabasco sauce and white wine. Simmer for 5 minutes to cook out the alcohol from the wine, stirring frequently to release the browned "bits" of chicken and sausage from the bottom of the pan.
Add sugar, tomatoes and chicken stock/broth; simmer for 30-45 minutes, or until liquid has reduced by at least half.
While sauce is simmering, cook pasta according to package directions. Do not overcook as pasta will continue cooking when added to the sauce.
When the sauce has reduced, remove and discard bay leaf (and fresh thyme stems if using), stir in half-n-half and then the chicken and sausage and return to a simmer. Make sure chicken is fully cooked.
Add pasta to the sauce and meat and toss until pasta is coated. Sprinkle with 1/4 cup of the Parmesan cheese and 1/4 cup of the green onions.
Serve with remaining Parmesan cheese and green onions.