Melt 2 tablespoons of butter in a large pot over medium heat. Add the diced onion and cook for 3 to 4 minutes until it softens and turns translucent. Stir in the garlic and let it cook for just one more minute so it doesn't burn.
Add the uncooked rice along with 1 tablespoon of Cajun seasoning directly into the pot. Stir everything together so each grain gets coated in that spiced butter. This little toasting step helps the rice hold onto more flavor as it cooks.
Pour in the chicken broth and bring it up to a boil. Once bubbling, drop the heat to low, cover with a tight lid, and let it simmer undisturbed for 20 to 25 minutes. Resist lifting the lid, since trapped steam is what cooks the rice evenly.
While the rice simmers, toss the chicken pieces with the remaining tablespoon of Cajun seasoning in a bowl. Melt the remaining butter in a large skillet over medium-high heat, then add the chicken. Cook for 8 to 10 minutes, flipping halfway, until golden and cooked through to 165°F.
In that same skillet, melt more butter and add garlic, cooking for about a minute. Stir in Cajun seasoning and heavy cream, scraping up the browned bits left from the chicken. Simmer for 3 to 5 minutes until the sauce thickens slightly and turns glossy.
Add the sauce and cooked chicken straight into the pot of finished rice. Stir gently until everything is coated and combined. Sprinkle fresh parsley over the top and serve it warm right away for the best texture.