Put oven rack in upper third of oven and preheat oven to 400°F.
Peel parsnips and cut crosswise into 3-inch lengths with a knife, then cut lengthwise around core into 1/8-inch-thick slices with slicer. Transfer to a large bowl.
Peel celery root and all potatoes (prepare potatoes last to avoid discoloration) and halve lengthwise, then cut crosswise into 1/8-inch-thick slices with slicer. Add to parsnips along with salt, garlic, pepper, nutmeg, broth, and 1 3/4 cups cream, tossing to combine. Transfer to gratin dish, spreading evenly.
Cut out a piece of parchment or wax paper to fit just inside gratin dish, then butter 1 side of parchment. Cover vegetables directly with parchment, buttered side down, then put dish in a shallow baking pan (to catch any drips). Bake until gratin is bubbling all over and vegetables are tender when pierced with a knife, about 50 minutes. Discard parchment.
Just before serving, preheat broiler. Drizzle top of gratin with remaining 2 tablespoons cream and broil 4 to 6 inches from heat until browned, 2 to 3 minutes.
Cooks' notes: Gratin, without final addition of cream (2 tablespoons), can be baked (but not broiled) 6 hours ahead and cooled completely, uncovered, then chilled, loosely covered. Reheat, covered, in a preheated 400°F oven until hot, about 30 minutes. Drizzle with cream and broil just before serving.
Adapted from Gourmet, November 2006