Heat butter in a large Dutch oven over medium-high heat. When butter is melted, add onions and garlic. Sauté for approximately 5 minutes or until aromatics are tender and fragrant, then stir in 1 tablespoon Cajun seasoning.
Remove pot from heat and stir in broth, diced tomatoes, and heavy cream. Return pot to heat and bring mixture to a boil, then reduce heat to medium and let liquids come to a rapid simmer.
Pour dry pasta into Dutch oven and stir well. Cook for 10 minutes, stirring regularly and making sure to scrape the bottom of the pot to prevent sticking or burning.
While pasta cooks, sprinkle remaining 1 tablespoon Cajun seasoning over chicken, covering all sides.
Heat a small skillet over medium heat. Once the pan is hot, pour in 1 tablespoon neutral oil and rotate the pan to cover the surface with oil.
Add seasoned chicken to skillet. Cook for 3 to 5 minutes, then flip chicken and cook on the other side for another 3 to 5 minutes. Transfer chicken to a cutting board and let cool to touch, then chop chicken into bite-sized pieces.
Reduce heat under Dutch oven to low. Remove pot from heat and stir in grated parmesan until well combined. Return pot to low heat and stir in chicken pieces, then let soup simmer for 2 to 3 minutes.
Taste soup and add salt and pepper to taste. Portion soup into serving bowls. Garnish with sliced green onions if desired and serve warm.