Preheat oven to 350 degrees.
Wash and clean the seafood. Cut seafood, snapper and monkfish into large pieces and place each type of seafood into individual baking pans.
Add the butter and olive oil to a large pot. Add garlic, onion, carrot, leek, and celery, and sweat until vegetables are soft. Deglaze the mixture with white wine and reduce the volume by half. Add cold water to the pot to cover the vegetables and heat mixture. Add the fish stock, chopped tomatoes, tomato paste, saffron threads, and freshly crushed black pepper to taste. Simmer for 90 minutes total and skim regularly. After 45 minutes add potatoes. After 90 minutes strain sauce, return potatoes and liquid to the pot and simmer until desired consistency. Correct seasoning to taste.
Cover each type of seafood in its individual baking pan with the sauce. Bake for 10 minutes or until seafood is cooked through and the mussels have opened. Discard any unopened mussels. Add seafood and sauce from each pan to bowls. Serve immediately.