Heat oven to 375 degrees. Rub a 2-quart gratin dish with garlic clove, and generously spread with butter. Set aside.
Melt butter in a small skillet over medium heat. Add breadcrumbs. Season with salt and pepper, and stir until crumbs have separated, 2 to 3 minutes. Transfer to a medium bowl, and stir in parsley. Set aside.
In a large bowl, combine pumpkin, thyme, and chives. Season with salt and pepper. Transfer pumpkin mixture to prepared dish.
In a small saucepan, combine cream and nutmeg. Bring to a boil. Reduce heat to medium; simmer 5 minutes. Pour cream over pumpkin mixture.
Transfer gratin dish to oven, and bake 20 minutes. Remove from oven; stir gently. Return to oven, and bake until cream has thickened and is bubbling, about 10 minutes. Remove from oven.
Turn on broiler. Sprinkle gratin with the reserved breadcrumb mixture, and broil until deep golden brown, about 2 minutes. Serve immediately.