Season the chicken: In a medium bowl, toss the diced chicken pieces with kosher salt, Italian seasoning, mustard powder, paprika, and chili flakes. Set aside.
Sauté the vegetables: In a large Dutch oven or soup pot, melt the butter over medium heat. Add the diced onions, carrots, and celery. Season with salt and sauté for 5-7 minutes until softened. Add the minced garlic and cook for an additional 30 seconds.
Cook the chicken: Add the seasoned chicken pieces to the pot and cook until no longer pink. Stir in the fresh thyme and rosemary.
Thicken the soup: Sprinkle the flour over the chicken and vegetables, stirring to coat evenly. Cook for 1-2 minutes.
Add the liquids: Slowly pour in the chicken broth, half and half, soy sauce, and hot sauce. Stir well. Add in the uncooked rice, stir again, and bring the mixture to a boil over medium-high heat. Reduce heat to low and simmer for 15-20 minutes, stirring frequently to prevent the rice from sticking to the bottom of the pot.
Adjust the consistency: If the soup becomes too thick, add additional broth or water to reach your desired consistency, especially when reheating leftovers.
Taste and serve: Adjust the seasoning with salt, pepper, and more hot sauce if desired. Ladle the soup into bowls and garnish with shaved Parmesan, freshly cracked black pepper, and fresh parsley. Enjoy!