Wash and dry the apples and set aside. Chop the hazelnuts and place them on a flat plate.
Heat the sugar in a medium saucepan over low heat, stirring constantly. The sugar will get chunky and then dissolve into a thick, brown amber-colored liquid. Be careful not to let it burn.
When the sugar has completely dissolved, immediately stir in the butter until melted and combined, and cook for 1 minute without stirring. Then very slowly, stir in the half-and-half. Since the cream is colder than the hot caramel, the mixture will bubble up quickly when you incorporate it, so be careful and add it slowly, stirring constantly. Cook for 2 minutes and remove from the heat.
Place the apples on popsicle sticks or skewers and, holding the end of the stick, dip the apple in the caramel and then coat with hazelnuts. Set aside on a plate lined with wax paper or parchment paper and continue with the remaining apples. Serve right away.
Notes
Tip: The waxy coating on the outside of apples can cause the caramel to slip off, so use hot water to wash it off as best you can.