Pour cider into small saucepan and simmer over medium heat until reduced to 1/3 cup; approximately 35 to 40 minutes. Set aside to cool.
Line 8-inch square pan with buttered parchment paper or aluminum foil. In small bowl, combine heavy cream, salt, apple pie spice, cinnamon and reduced cider; set aside.
In large, heavy-bottom saucepan, combine sugar and corn syrup. Cook over low heat, stirring until sugar dissolves. Insert candy thermometer and simmer until syrup reaches 234°F
Remove from heat, remove thermometer and very slowly whisk in cream mixture (mixture will foam and may spatter). Add cubed butter and stir or whisk until cream and butter are fully incorporated. Return pan to heat, reinsert thermometer and cook over low heat, stirring frequently, until temperature reaches 250°F
Remove from heat and pour caramel into prepared pan; let mixture cool slightly, then sprinkle with freshly ground cinnamon or sea salt, if desired.
Cool completely at room temperature or refrigerate until set. Once set, remove caramel from pan lifting out by parchment paper or foil. Cut caramel into 3/4-inch squares. Wrap each piece in small wax paper square, twisting at each end. Store caramels in airtight container in cool place or refrigerate up to 2 weeks.