Soak the cod in cold water in the refrigerator for 24 hours, changing the water 4 times.
Drain the cod. Put in a saucepan, cover with water and bring to a bare simmer; drain, then repeat with fresh water. Drain again, then add the milk and cream to the pan. Bring to a simmer over medium-low heat and cook until the fish is flaky, about 15 minutes. Reserve 1/2 cup of the milk mixture, then drain and coarsely flake the fish.
Put the potatoes in a saucepan and cover with cold water. Bring to a simmer over medium heat and cook until tender, about 20 minutes; drain. Let cool slightly, then press through a ricer or a food mill. Meanwhile, combine the olive oil, garlic, thyme and cloves in a small saucepan; cook over low heat, 5 minutes. Strain the flavored oil and set aside.
Put the cod, half of the flavored oil, the reserved milk mixture, the lemon juice and 1/2 cup hot water in a food processor; pulse until smooth. Add the potatoes and the remaining flavored oil and pulse to combine; season with pepper. Transfer to a bowl and drizzle with olive oil. Serve warm with crackers.