Butter a 2-quart casserole dish and set aside. Cut potatoes crosswise into 1/2-inch-thick slices.
Bring a large (5- to 6-quart) pot to a boil over high heat and add sweet potatoes. Cook until mostly tender, 10-15 minutes.
Meanwhile, core and slice apples into 1/2-inch-thick wedges. Preheat oven to 350°.
Drain potatoes in a colander. In a 2- to 3-quart pot, melt butter over medium-high heat; add cider, brown sugar, nutmeg, and allspice. Simmer until somewhat thickened, about 10 minutes. Set aside 1/4 cup syrup.
Arrange alternating layers of potatoes and apples in casserole dish; drizzle remaining syrup over each layer.
Cover with foil and bake until tender, about 20 minutes. Remove foil, and brush on reserved syrup. Set broiler to high; broil casserole until nicely caramelized, 3-5 minutes.