Cut the pork and liver into 2 inch pieces and place in a Dutch Oven, along with the onion, garlic, thyme, and bay leaves.
Cover with cold water by 2 inches. Season the water well with salt and black pepper. Bring to a boil then lower the heat to a simmer, skim off any scum that rises to the surface. Simmer for about 1 hour or until the meat is very tender. Remove the bay leaves, and thyme, then strain the solids from the broth, reserving some of the broth.
Run the cooked meat and vegetables (while they’re still hot) through a meat grinder or food mill, or chop by hand.
Combine the cooked rice with the ground meat mixture, green onions, and parsley. Mix thoroughly and season to taste with salt, black pepper, and cayenne. Add some of the reserved cooking liquid to make sure that the finished product is very moist, bearing in mind that the rice will continue to absorb much of the liquid.
Spread the mixture on a sheet pan and place in the refrigerator to cool.
When the mixture is cool form into patties or balls for pan-frying. Boudin also makes a great stuffing.