Cook the pasta in salted water according to the package directions until al dente. Drain the pasta, reserving 1 cup cooking water; keep warm.
Heat 1 tablespoon of the oil in a large Dutch oven over medium-high heat. Sprinkle the shrimp with 2 teaspoons of the Cajun seasoning, and add to the Dutch oven. Cook, stirring occasionally, just until the shrimp turn pink, about 3 minutes. Transfer to a plate. Add 1 tablespoon of the oil to the Dutch oven; sprinkle the chicken with 2 teaspoons of the Cajun seasoning. Add the chicken to the Dutch oven, and cook, turning to brown on all sides, until done, about 5 minutes. Transfer the chicken to the plate with the shrimp. Add the remaining 1 tablespoon oil to the Dutch oven; add the sausage, onion, and bell pepper. Cook, stirring often, until the sausage is browned and the vegetables are tender, about 5 minutes. Add the garlic; cook 1 minute. Add the stock, and cook 30 seconds, stirring and scraping to loosen browned bits from the bottom of the Dutch oven. Add the tomatoes, thyme, and remaining 2 teaspoons Cajun seasoning; cook about 2 minutes. Stir in the heavy cream.
Add the shrimp, chicken, pasta, and reserved 1 cup cooking water to the Dutch oven; toss to combine. Cook until the mixture is thoroughly heated and the sauce thickens, 1 to 2 minutes. Remove from the heat, and add the Parmesan and basil; toss to combine.
Recipe adapted from Southern Living's "What Can I Bring?" by Elizabeth Heiskell. Reprinted with permission from Meredith Corporation.