1. Melt the butter in a saucepan. Add the brown sugar and water. Stir until it has a smooth consistency. Gently bring to a boil, then cover and simmer for about 3 minutes, until the pan's sides get steamy and the mixture is thin and sticky. Remove from heat.
2. Pierce the center of each apple with a Popsicle stick, and then swirl in caramel syrup until coated. Place apples on a greased cookie sheet to harden.