Heat the oil in a heavy 9 1/2-inch-diameter skillet over medium heat. Add the potato, onion, and garlic. Season the potato mixture with salt and pepper, to taste. Saute over medium-low heat until the potato is golden and crisp on the outside, tender inside, about 7 to 10 minutes.

Preheat the broiler. Whisk the eggs, cream, Parmesan, prosciutto, and basil in a medium bowl to blend. Stir the egg mixture into the potato mixture in the skillet. Cook over medium-low heat until the egg mixture is almost set but the top is still loose, about 3 minutes. Place the skillet under the broiler. Broil until the top is set and golden brown, about 4 minutes. Using a rubber spatula, loosen the frittata from skillet and slide the frittata onto a cutting board. Cut the frittata into 12 wedges.

Split open each bread piece. Toast the bread on a grill pan until golden. You can also broil the bread cut side up until golden, about 1 to 2 minutes, keeping a careful watch on the bread since they can easily burn. Butter the cut sides of the toasted bread. Arrange a frittata wedge on each bottom piece of bread. Cover with the bread tops and serve.