Soak the salt cod in water overnight (24 hours), changing the water every 4-8 hours or so. Strain the water completely. After 24 hours, bring the pint of milk and pint of water with the bay leaves and the hydrated cod to a boil. Reduce the heat to a simmer and cook slowly for 1 hour. Gently strain the mixture and reserve all of the fish. Cook the diced potatoes in boiling salted water for 3-4 minutes. Remove them from the flame and rinse with cold water. Reserve for later.
Slice the fresh garlic. Strain the tomatoes and cut them into eighths. Heat a large skillet or casserole over medium flame for 2-3 minutes. Add ½ of the olive oil and heat 20-30 seconds. Add the sliced garlic and cook in the oil until just very lightly browned. Add the cut tomatoes to the garlic and sauté for 2-3 minutes. Add the cooked salt cod and the olives and simmer the mixture for 20 minutes, covered. Add the potatoes and continue to simmer for another 15-20 minutes.
Serve this dish warm or cold with more extra-virgin olive oil and fresh basil. Season the casserole with salt and pepper to taste.