Heat the oil over medium-high heat in a large, heavy pot or Dutch oven. Add the rabbit and cook until lightly browned, about 5 minutes on each side. Transfer the rabbit to a plate lined with paper towels. Reduce the heat to medium, add the flour, and stir constantly to make a golden-brown roux, about 10 minutes. Add the onions, celery, bell peppers, white parts of the green onions, and the garlic. Cook, stirring, until the vegetables are very soft, about 10 minutes. Stir in the broth and bring to a boil. Return the rabbit to the skillet and simmer, covered, until the rabbit is very tender, about 3 hours, adding more broth as necessary if the mixture becomes too dry.
Add green parts of the green onions to rabbit and cook for 15 minutes. Serve the rabbit and gravy over rice.
Makes 4 servings