To make the apple caramel sauce,add half of the Martinelli’s apple juice and sugar to a medium saucepan. Over medium-high heat, cook the liquid until it is dark brown in color.
Add the heavy cream, butter and remaining Martinelli’s apple juice. Be careful as the mixture will bubble up! Stir well and let cool completely.
To make the cakes, preheat oven to 425 degrees F. Butter and flour 4 ramekins; add them to a rimmed baking sheet and set aside.
In the bowl of a stand mixer, beat egg yolks and egg in a medium bowl until the mixture has doubled in size. It will be voluminous and will hold a slight ribbon.
Add the apple caramel sauce to the mixer, beating until well blended. Gradually add flour, beating until well blended. Divide the batter between the 4 ramekins.
Add sheet pan to the oven. Bake until the outsides are golden brown and the centers still jiggle, approximately 11 minutes.
Remove from oven and run a knife around the edges of the ramekins to loosen cakes. Invert cakes onto plates. Serve immediately with vanilla ice cream.