In a deep heavy saucepan combine the brown sugar, maple syrup, heavy cream and cook the mixture over moderate heat until the sugar is dissolved. Wash down any sugar crystals that cling to the sides of the pan with a brush dipped in cold water.
Bring the mixture to a boil and let it cook, undisturbed, until a candy thermometer registers 250°F. Remove the pan from the heat, add the butter and swirl the pan gently until the butter is melted. Let the maple caramel cool for 5 minutes.
Push a popsicle stick or chopstick through the length of each apple and holding the apples by the sticks dip the apples 1 at a time in the maple caramel, coating them well and letting the excess drip off. Put the apples on buttered baking sheets until the maple caramel is cool and firm.