In a large Dutch oven, heat the olive oil (or the oil and butter mixture) over medium heat. Add the chopped onion, celery, and carrots. Sauté for about 5-7 minutes, or until the vegetables are softened and the onions are translucent.
Stir in the minced garlic and cook for an additional minute, allowing the garlic to become fragrant but not burnt.
Push the vegetables to the sides of the pot and add the chicken pieces, skin-side down. Let them brown for about 5-6 minutes without moving them. Flip the chicken and brown the other side for a few minutes.
Once the chicken is browned, add the long-grain rice, fresh thyme, and ground cumin to the pot. Stir everything together, ensuring the rice is well-coated with the oil and spices.
Pour in enough water or chicken broth to cover the chicken and rice mixture by about an inch. Season generously with salt and pepper. Bring to a boil, then reduce the heat to low and cover. Simmer for about 30-35 minutes, or until the chicken is cooked through and the rice is tender.
Once cooked, remove the pot from heat and let it sit, covered, for 5 minutes. Before serving, sprinkle chopped parsley over the top for a fresh touch.