Line a baking sheet with parchment. Stem, wash and dry apples and insert Popsicle sticks in center of stem ends. Fill a large bowl halfway with ice cubes and water.
Combine sugar, corn syrup, cream and butter in a medium-size, heavy-bottom saucepan. Cook over low heat, gently stirring once or twice, until sugar has dissolved. Increase heat to medium-high and cook, stirring often, until mixture reaches 255ºF on a candy thermometer, about 20 minutes.
Set pan in bowl of ice water. Stir just until mixture starts to thicken into a dipping consistency. Working carefully, tip pan slightly to one side and spin each apple in caramel until thoroughly coated. (If mixture becomes too thick, reheat gently for a few seconds.) Transfer to lined baking sheet and place in refrigerator until caramel is cooled and set, about 15 minutes.