Finely dice the onion, celery, and green pepper and set aside.
Rinse the chicken livers and dice them fine, removing any membranes. Set aside.
Heat the oil in a dutch oven or large skillet over medium heat.
When the oil has sufficiently heated add the pork. Cook until browned, breaking up the pork into small bits, until browned.
Add the celery, onion, and green bell pepper to the pan. Cook until the vegetables are softened, about 10 minutes. Stir the vegetables occasionally.
Add the chicken livers, kosher salt, black pepper, garlic, thyme, and cayenne pepper to the pan and continue to cook for about 5 minutes, until the livers are browned.
Transfer the meat mixture to a strainer set over a bowl and allow it to drain. Cover with aluminum foil to retain some heat while the rice is being prepared.
Increase the heat to medium-high to high heat and add the chicken broth, rice, and bay leaves.
Bring to a boil and scrape the pan of any browned bits on the bottom of the pan with a wooden spoon.
Once it reaches a boil, reduce the heat to a simmer and cover. Cook until the rice is tender, about 17 minutes.
Remove from the heat and remove the bay leaves. Fluff up the rice with a fork then add the meat mixture and combine well.
Garnish with some chopped green onions and serve immediately.