To a medium saucepan combine the peeled and diced apples, apple juice, lemon juice, brown sugar, cornstarch, cinnamon and nutmeg.
Stir to combine, and bring to a simmer over medium heat, stirring often.
Let the mixture simmer for 8-10 minutes, or until the mixture thickens. Stir every 2-3 minutes, scraping down the sides of the saucepan.
Once thickened, remove the saucepan from heat, and pour the mixture through a strainer, into a heatproof container. Place the apple chunks in a separate container.
Let the mixture cool at room temperature for at least one hour, then transfer to an airtight container and store in the refrigerator for up to 3 weeks.