Soften ice cream by leaving at room temperature for 20 minutes or until soft enough to mix. Transfer to a large mixing bowl. Using an electric mixer, beat ice cream and cinnamon until thoroughly combined. Cover and return to freezer for 30 minutes or until solid. (Can be done up to a week in advance.)
Preheat oven to 350°F. Butter a baking dish just large enough to hold apples in a single layer.
Combine sugar, cider, salt, nutmeg and cloves in a small saucepan. Bring to a boil over high heat, stirring until sugar has dissolved. Reduce heat to medium and cook for 5 minutes. Remove from heat.
Use a vegetable peeler to trim about 1/2-inch band around tops of apples. Scoop out cores and seeds to form a deep cavity, without piercing bottom, using an apple corer, a melon baller or a sharp spoon. Place apples upright in baking pan. Drop a few bits of butter in each apple cavity and pour cider syrup over and around the apples.
Bake apples, uncovered, basting often, until apples are tender but still hold their shape, 45 to 50 minutes. Cool slightly before serving with extra cooking juices spooned over and cinnamon ice cream on the side.