In a bowl, toss the chicken with 1/2 tsp of the salt, Italian seasoning, ground mustard, smoked paprika, and red pepper flakes until evenly coated.
Warm a large Dutch oven over medium-high heat. Add the olive oil and 1 tbsp butter. Add the seasoned chicken in an even layer and cook, stirring once or twice, until lightly browned and no longer pink on the outside, 4–6 minutes. Transfer to a plate (it will finish cooking later).
Reduce the heat to medium and melt the remaining 1 tbsp butter. Add the onion, carrots, and celery with a pinch of salt. Cook, stirring occasionally, until the vegetables are softened and starting to turn sweet, 6–8 minutes. Stir in the garlic and cook 30–45 seconds until fragrant.
Sprinkle the flour over the vegetables and stir constantly for 1–2 minutes to form a light roux.
Slowly pour in about 1 cup of the chicken stock, stirring to make a smooth, thick base and scraping up any browned bits. Add the remaining stock, soy sauce, and hot sauce. Return the chicken and any collected juices to the pot and bring just to a simmer.
Stir in the rice, reduce the heat to low, and simmer gently, partially covered, until the rice is tender and the soup has thickened, 15–18 minutes. Stir every few minutes to keep the rice from sticking to the bottom.
Pour in the half-and-half and warm through 3–5 minutes more. If the soup seems too thick, loosen with a splash of stock or water; if too thin, simmer a couple more minutes. Season with the remaining 1/2 tsp salt and several grinds of black pepper to taste.
Let the pot rest off the heat for 5 minutes to settle. Ladle into bowls and finish with Parmesan, parsley, and extra black pepper.