Add the oil to a Dutch oven on medium-high heat. When oil is heated, add the chicken pieces skin side down. Sprinkle on some salt and pepper and cook for about 5-10 minutes.
Turn the chicken over and sprinkle on a little more salt and pepper (to taste) and cook; approximately 10-15 minutes total. When evenly browned, remove the chicken pieces from the pan.
Remove most of the oil (you'll just need 2 Tablespoons) and add the onions. Cook until they are transluscent. Add the rice, saffron and bay leaf. Stir to combine.
Add the broth and bring to a boil. Place the chicken pieces back in the pan and make sure the rice is surrounding the chicken all around it (not all the rice in one pile underneath it).
Cover the pan with the lid. Lower the heat and let it simmer for about 25 minutes. (If you would like to check the temperature of the chicken, put the thermometer in thickest part of the thigh and temperature should be between 155-165 F) The rice should be tender and the chicken cooked through.
If the rice seems too dry and the chicken isn't ready, add another 1/4 cup of boiling stock and let it cook for about another 10 minutes.
When it is cooked, remove the pan from the heat; add the peas and check the seasoning. Add the lime juice (optional).
Add more salt and pepper, if needed, according to your personal taste.
Remove the bay leaf.
Stir together to combine and let the pan sit with the lid on for about 10 minutes.