Over medium heat, melt butter in a 6-quart Dutch oven.
Add the sausage and ham and cook until all fat is rendered, about 5 minutes, stirring occasionally.
Add the yellow onions, the white part of the green onions, and the peppers. Cook the veggies until they're clear, about 5 minutes, stirring occasionally and scraping the pan bottom clean.
Add the crushed tomatoes, tomato puree, garlic, bay leaf, salt, black pepper, cayenne pepper, and thyme. Cook and stir about 2 minutes.
Add the chicken stock and pepper sauce and bring to a boil.
Reduce the heat to maintain a strong simmer, and simmer uncovered until it is reduced by 1/3, about 1.5 hours. Skim any foam or coagulates as they develop on the surface.
Return the liquid to a boil and stir in the rice.
Reduce the heat to medium and cook uncovered until the rice is almost done, about 25 minutes, stirring occasionally.
Add the shrimp and cook until the rice is tender and the shrimp are bright pink, about 3 minutes.
Stir in the green part of the green onions and serve.
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