This is a savory side dish that is baked in a creamy Gruyere sauce.
Preheat oven to 425 degrees with rack positioned in the center. Squeeze the juice from two lemon halves into a medium bowl of water. Peel and cut the Jerusalem artichokes lengthwise into thirds, and add to the acidulated water. Butter a shallow 7 1/2-by-11 1/2-inch ovenproof baking dish, and set aside.
Place the potatoes, carrots, garlic, and 2 tablespoons parsley in a large bowl. Remove the Jerusalem artichokes from the water, pat dry, and add to the bowl. Sprinkle generously with salt and pepper. Toss to combine, and arrange the vegetables in an even layer in the prepared dish. Dot with the butter.
Pour cream into a small saucepan; heat over medium-high heat just until it comes to a boil. Pour hot cream over vegetables; cover dish tightly with aluminum foil. Place the dish on a rimmed baking sheet to catch any cream that might bubble over. Transfer both baking sheet and dish to oven, and cook until potatoes begin to soften, about 30 minutes.
Reduce oven temperature to 400 degrees. Remove aluminum foil; stir vegetables so they are coated with cream. Continue cooking until the tip of a knife easily pierces potatoes and artichokes, about 30 minutes more.
Preheat the broiler. Using the large holes of a box grater, grate the cheese, and sprinkle over the gratin. Place dish under broiler, and cook just until the cheese is golden brown and crusty. Sprinkle with remaining tablespoon parsley, and serve hot.
Jerusalem artichokes are nutty, slightly sweet tubers that are members of the sunflower family.