Cut the "beards" off the mussels. Scrub well and soak in several changes of cold water. Discard any with broken shells, Place the mussels with a cupful of water in a large saucepan, covered, over medium heat. As soon as they open, lift them out, (Discard any that do not open.) Remove and discard the empty half-shells, leaving the mussels in the other half. Strain any liquid in the pan through a layer of paper towels and reserve.

Boil the potatoes in salted water until they are almost cooked but still firm, then peel and slice them thinly,

Preheat the oven to 350°F. Spread 2 tablespoons of the olive oil in the bottom of a shallow ovenproof dish. Cover with the potato slices in one layer. Add the mussels in their half-shells in one layer. Sprinkle with garlic and pieces of basil.

Cover with a layer of the tomato slices. Sprinkle with bread crumbs and salt and pepper, the strained mussel liquid and the remaining olive oil. Bake for about 20 minutes, or until the tomatoes are soft and the bread crumbs are golden. Serve directly from the baking dish.