In France, this is served with sliced black truffles, but truffle oil is a good substitute.
Soak salt cod in cold water in the refrigerator 12 hours, changing the water three or more times. When ready, the cod should taste a bit saltier than a fish that you’ve seasoned and cooked with salt.
Preheat oven to 450 degrees F. Bake potato until tender, about 50 to 60 minutes. While the potato is cooking, drain and rinse salt cod, then place in a medium pot covered with cold salted water. Bring to a boil. Reduce heat; simmer until fish is cooked through, about 10 minutes. Set aside to cool in liquid. As soon as the potato is done, split it in half, scoop out the steaming flesh, and pass through a ricer.
Heat olive oil in a small saucepan over moderate heat. Add garlic and shallots; cook until soft, then add cream and simmer 7 minutes. Remove from heat. Drain cod and pat dry. Remove the skin and bones, then pulse a few times in food processor to break into coarse pieces. With the machine running, add the garlic-cream mixture in a steady stream. Transfer to a medium bowl, add the potato, and stir until everything is mixed well. Season with salt, pepper, and 1 tablespoon lemon juice. Add more salt and/or lemon juice if necessary. If not using the brandade immediately, cover and refrigerate. (It will keep for up to a week.)
To serve, place in a large saute pan and warm over medium heat, stirring constantly, until heated through. Stir in parsley, spoon into a serving bowl, and serve surrounded by toasts. Garnish with a few olives and drizzle with truffle oil if you have it.