Prepare the apples by coring, peeling, and dicing them. To prevent oxidation, coat the apples in fresh lemon juice and set aside.
In a large saucepan, combine the brown sugar, granulated sugar, cornstarch, cinnamon, and nutmeg. Add water and whisk until all ingredients are dissolved. Bring the mixture to a boil over medium-high heat.
After reducing the heat slightly, allow the mixture to simmer for 2 minutes. Add the apples, return to a boil over medium-high heat, and then reduce to low. Simmer for 10-12 minutes, or until the apples are softened and the sauce has thickened.
Remove from heat and allow the mixture to cool to room temperature.