In a wide, heavy-bottomed, deep-sided pan, melt sugar over medium heat, stirring once or twice in the beginning once the edges begin to melt.
When it darkens to your desired color (we like it to be the color of a copper penny -- this will happen just after it starts to smoke), stir in butter and cream. It will bubble up at first, but will calm down quickly as you stir. Remove from heat.
Skewer each apple with a popsicle stick and roll in warm caramel to coat, dipping twice for extra-thick caramel. Place on parchment-lined tray to cool. Serve once the caramel is firm, or refrigerate for up to 3 days.