Heat olive oil in a large pot or Dutch oven over medium heat. Add the diced onion, garlic, bell pepper, carrots, and celery. Sauté for about 5-7 minutes until the vegetables are softened.
Season the chicken thighs with salt, pepper, and Cajun seasoning. Add the chicken to the pot and cook for about 5-6 minutes, stirring occasionally, until the chicken is browned on all sides.
Pour in the chicken broth and the can of diced tomatoes, bringing the mixture to a boil. Reduce the heat to a simmer.
Add the uncooked pasta and dried thyme to the pot. Continue to simmer for about 10-12 minutes, until the pasta is cooked al dente.
Stir in the heavy cream and adjust the seasoning with additional salt, pepper, or more Cajun seasoning if desired. Let the soup simmer for an additional 2-3 minutes.
Remove the chicken from the pot, shred it with two forks, and return it to the soup. Stir well to combine.
Serve hot, garnished with sliced green onions and chopped fresh parsley.