In a large, heavy pot or Dutch oven heat the oil over medium-high heat. Season the rabbit pieces with the Essence and, working in batches if necessary, cook until browned on all sides, about 10 minutes. Transfer the rabbit to a plate lined with paper towels and reduce the heat to medium. Add the flour to the oil in the pot and whisk to combine. Cook, stirring constantly, until the roux is a golden brown color, about 10 minutes. Add the onions, celery, bell pepper, tomatoes, white parts of the green onions and the garlic and cook until vegetables are soft, about 10 minutes. Stir in the tomato paste and chicken stock and bring to a boil. Return the rabbit to the pot and return to a boil. Add the thyme and salt, reduce heat to a simmer, and cook, uncovered, until the rabbit is very tender, about 2 hours, adding more broth or water, if necessary, if the sauce gets too thick.

When the rabbit is tender and you are ready to serve the stew, add the green parts of the green onions and parsley and cook for 5 minutes. Serve the rabbit with the gravy over hot white rice.