Make the caramel: Pour the sugar into the center of a deep saucepan. Carefully pour the water around the walls of the pan, trying not to splash any sugar onto the walls. Do not stir; gently draw your finger twice through the center of the sugar, making a cross, to moisten it. Over high heat, bring to a full boil and cook without stirring until amber caramel in color, 5 to 10 minutes. Immediately turn off the heat and use a wooden spoon to stir in the butter. Slowly stir in the cream (it will bubble up and may splatter). Let cool slightly to thicken.
Pour the caramel onto a well-buttered sheet pan and let cool to set up, approximately 2 hours. Cut walnut size squares of it and roll it in your hands into a ball then shove a sucker stick in the center and roll it in chopped peanuts. Place them on a plate covered with peanuts and chill 30 minutes to make them set up more.