Peel, core, and slice the apples into thin wedges or bite-sized pieces. To prevent browning, toss the slices with lemon juice immediately after slicing.
Melt the butter over medium heat in a large skillet or sauté pan.
Add the granulated sugar, brown sugar, salt, ground cinnamon, and ground nutmeg (if using).
Stir the mixture until the sugars dissolve and the butter melts. Stir until well combined and a syrupy glaze forms. It should take 2-3 minutes.
If you prefer a thicker sauce, mix the cornstarch with a small amount of water to create a slurry.
Stir the slurry into the pan and cook for 2-3 minutes or until the sauce reaches your desired consistency.
Add the prepared apple slices and vanilla to the pan, tossing them gently to coat them in the syrupy mixture.
Continue to cook, stirring occasionally, for 7-10 minutes or until the apples are tender and the sauce has thickened slightly. The cooking time may vary based on the thickness of the apple slices.
Remove the skillet from the heat. Taste and adjust the seasoning if necessary.