Rinse the cod under cold running water and put it in a large bowl containing about 5 quarts cold water. Let soak for at least 4 hours, changing the water after 2 hours.

Meanwhile, put the potatoes in a saucepan, cover them with cold water, and bring to a boil, then reduce the heat and boil the potatoes gently for about 30 minutes, until tender. Drain and set aside.

Drain the cod. Put it in a saucepan with 8 cups cold water, bring just to a gentle boil, and drain immediately. Rinse out the pan and add the cod and 4 cups cold water. Bring to a boil and boil for 5 minutes. Drain.

When the cod is cool enough to handle, remove any skin and bones with your fingers, breaking the flesh into pieces. Return the cod to the pan, add the milk and garlic, and bring to a boil, then cover, reduce the heat to low, and cook gently for 10 minutes.

Peel the potatoes and cut them into 2-inch pieces.

Preheat the oven to 400 degrees.

Put the cod, with the milk, and the garlic, in a food processor. Add the lemon zest and potatoes and process for about 1 minute, until smooth. Add the lemon juice, black pepper, and cayenne pepper and process briefly to incorporate. With the motor running, slowly add the oil and continue processing until the mixture is very smooth.

Lightly oil an 8-cup gratin dish and transfer the brandade to the dish, spreading it out evenly. (The brandade can be made several hours ahead. Cover with plastic wrap and refrigerate.)

Sprinkle the brandade with the cheese and bake for 10 to 15 minutes, until the top is lightly browned and it is hot throughout. (If the brandade has been refrigerated, increase the baking time by about 10 minutes.)

Meanwhile, arrange the rounds of bread in a single layer on a cookie sheet. Bake for about 10 minutes, or until nicely browned.

Serve the brandade with the toasts for dipping or spreading.