Prepare. If you have not done so, peel your apples, remove the seeds, and then cut the apples into one to two inch cubes or quarter inch slices. Then place the apples into a large bowl.
Toss. In a small bowl, combine together the sugars, ground cinnamon, nutmeg, and lemon juice. Pour half of this mixture over the apples and toss the ingredients together. Repeat this process with the remaining mixture, tossing the apples so that they are are all coated with the ingredients.
Heat. In a large skillet on medium heat, melt the butter. When the butter has melted add the ingredients from the bowl and cook the ingredients until the apples become softened and the sugar has completely dissolved, stirring occasionally (about 10-12 minutes to soften).
Slurry. While your apples are cooking you can make your slurry. Whisk together the cornstarch and water in a small bowl. When the apples have softened from the last step turn the heat on the stove down to low and add the slurry to the apples, stirring constantly, until the liquid has thickened and is sticking with the apples (about two to four minutes).
Cool. Allow the mixture to completely cool before transferring to a jar for storage.