Pour the apple juice into a small saucepan and bring it to a gentle simmer.
Simmer for 10–15 minutes, or until the apple juice has reduced to about 3/4 cup.
Add the brown sugar, granulated sugar, cinnamon, nutmeg, and salt. Reduce the heat to low and stir until the sugars are completely dissolved and the syrup is smooth.
Remove the saucepan from the heat and allow the syrup to cool for a few minutes.
Stir in the vanilla extract.
Allow the syrup to cool before pouring it into a clean syrup bottle or jar. Store in the refrigerator.