Place sweet potatoes into a large pot and cover with water; bring to a boil. Reduce heat to medium-low and simmer until tender, about 20 minutes; drain. Transfer drained sweet potatoes to a baking sheet.
Bake sweet potatoes in the preheated oven until dried, 3 to 4 minutes; transfer to a mixing bowl.
Mash sweet potatoes with a potato masher until no large lumps remain; add sugar, allspice, and cinnamon. Continue mashing potatoes until evenly seasoned; spoon into a casserole dish. Spread miniature marshmallows in an even layer over the sweet potato mixture.
Bake in oven until marshmallows just begin to brown, 10 to 15 minutes.
Switch oven to broil; cook under broiler for 1 minute.