Step 1

Put the salt cod in a bowl and cover with 1 inch of cold water. Refrigerate for 24 hours, changing the water 4 times.

Step 2

Put the potatoes in a large saucepan, cover with water and boil over moderately high heat until tender, about 30 minutes. Drain and let the potatoes cool to warm.

Step 3

Meanwhile, drain the cod and transfer to a saucepan. Add 2 quarts of water; bring just to a boil. Drain the cod and rinse out the pan. Return the cod to the pan, add 4 cups of water and bring to a boil. Simmer over low heat for 5 minutes. Drain the cod and return it to the pan, add the milk and garlic and bring to a boil. Cover and simmer for 10 minutes.

Step 4

Peel the potatoes and break into chunks; transfer to a food processor. Add the cod, milk, garlic cloves, lemon zest, lemon juice and cayenne and process until smooth. With the machine on, slowly pour in the 3/4 cup of oil until incorporated. Season with black pepper.

Step 5

Preheat the oven to 400°. Lightly oil a 9-by-13-inch baking dish and spread the brandade in the dish. Sprinkle the cheese on top. Bake on the top shelf of the oven for 20 minutes, until golden brown. Serve with toasts.