In a large saucepan, cover potatoes and garlic with cool water. Season with salt and bring water to a boil over high heat. Reduce heat to low, cover saucepan and simmer until potatoes are tender when pierced with a fork, about 15 minutes.
Drain potatoes and garlic and return to pot. Heat over high heat, shaking pan, until any liquid has evaporated, about 30 seconds. Mash potatoes and garlic until smooth. Add cream, chives and butter; season with salt and pepper. Transfer potato mixture to a shallow 2-quart flameproof baking dish. (The gratin can be prepared to this point up to 5 hours ahead. Cover and keep at room temperature.)
Preheat broiler. Sprinkle potatoes with Parmesan. Broil gratin 2 inches from heat, rotating pan if necessary, until golden, crusty and heated through, 5 to 7 minutes.