In a large bowl, combine the sliced onions, minced garlic, lemon juice, vegetable oil, Dijon mustard, white vinegar, salt, black pepper, cayenne pepper (if using), and bay leaves.
Add the chicken pieces to the bowl, ensuring they are well-coated with the marinade.
Cover and refrigerate for at least 2 hours, preferably overnight, to allow the flavors to meld.
Remove the chicken pieces from the marinade, reserving the marinade and onions.
In a large pot or Dutch oven, heat the vegetable oil over medium-high heat.
Add the chicken pieces and brown on all sides. Remove the chicken and set aside.
In the same pot, add the reserved marinade and onions. Sauté until the onions are caramelized and softened, about 10-15 minutes.
Add the chicken broth or water to the pot and bring to a simmer.
Return the browned chicken to the pot.
If using, add the sliced carrot, bell pepper, and green olives to the pot.
Cover and simmer on low heat for about 30-40 minutes, or until the chicken is cooked through and tender. Stir occasionally to prevent sticking and ensure even cooking.
Remove the bay leaves.
Enjoy your meal!