Hold the carrot firmly in your hands or place it on a flat surface. Run peeler from top to bottom of the carrot pressing firmly until you have even cut slender sticks. Let the carrot sticks drop into a large bowl.
Place a small skillet on a medium heat. Add sesame seeds and toast them for a minute or two until they start browning. Make sure not to burn the seeds. Remove from the heat and transfer the sesame seeds to a small bowl. Set aside.
Preheat the oven to 400 F. Stir the apples into the caramel sauce and cook until the apples caramelize turning them every few minutes to coat in caramel. Add in vanilla extract, stir once again and remove apples from the heat.