In a small bowl make a marinade by combining and whisking oil, and the seasonings. Coat the chicken pieces evenly with the marinade, and let rest between 5-30 minutes. (If it's hot out leave it in the fridge.) Heat oven to 485°F (250°C). Grease the pan with oil and place the drumsticks and thighs in. Also, add the remaining marinade. Bake for 15 minutes.
Take the pan out, and flip the chicken pieces around. Return the pan to the oven. (If using glassware there is no need to flip them around.) Lower heat to 360°-400°F (180°-200°C), depending on how hot your oven gets. Bake for another 15 minutes.
Take the pan out of the oven. Then take the chicken pieces out of the pan and set them aside. Carefully scrub any chicken parts and grease remaining in the pan after baking, and leave them inside the pan.
Add rice and 4.5 cups of water (or chicken broth). Stir it around so it's evenly distributed. (Also add potato and carrots if using them.) Return the pan to the oven, bring the heat up to 400-420°F (200°-210°C), and bake for about 25 minutes.
At the end of baking, or near the end (when there is half an inch of liquid left in the pan above the rice), add chicken on top of the rice. If there is still a lot of liquid left, leave the oven on for another 10 minutes. Alternatively, turn the oven off and leave the pan in for another 10 minutes. (If you like your rice drier, keep it in a little longer.)
Serving: serve warm, with homemade bread and a side salad. Storing: Warning - this dish tastes best immediately after baking and up to a few hours later. (Chicken made this way dries out after this period.) Keep baked chicken in the turned-off oven for a couple of hours. Later transfer to the fridge for up to 24 hours, in a closed container. Reheating: reheat using your preferred method.