How to Make It
Preheat the broiler. In a large saucepan of boiling salted water, cook the potatoes until just tender, about 20 minutes. Drain and let cool, then cut each potato into neat 1/3-inch-thick slices.
On an oiled baking sheet, arrange each potato into a round of slightly overlapping slices. Sprinkle each round with a little of the shallot and 2 teaspoons of olive oil and season with salt and pepper. Arrange 5 tarragon leaves on each round and top with 1 slice of cheese. Broil the potatoes, rotating the pan, for about 2 minutes, or until the cheese is bubbly but not browned.
Carefully transfer the potato rounds to plates using a metal spatula. Sprinkle 1/2 teaspoon of the balsamic vinegar around the potato on each plate and serve at once.