Heat the oil in a large saucepan over medium-high heat. Saute the onions, bell peppers, celery, salt, cayenne, black pepper and thyme for about 5 minutes. Add the bay leaves, ham and sausage and saute for 5 to 6 minutes. Add the beans, garlic, and enough water to cover the contents in the pot. Bring to a boil, reduce the heat to medium, and simmer uncovered, stirring occasionally for about 2 hours. Add more water if the mixture becomes dry and thick. Use a wooden spoon to mash about half of the mixture against the side pf the pot. Continue to cook, stirring occasionally, for about 1 1/2 hours, or until the mixture is creamy and the beans are soft. Add more water if it becomes too thick. The mixture should be soupy but not watery. Remove the bay leaves and serve with steamed white rice.
Recipe from Emeril Lagasse