Peel the skin from the apples, remove the core, then cut them into slices. Optionally, squeeze lemon juice onto the apple slices to keep them from browning.
Heat a frying pan over medium-low, and melt the butter. Then, add the brown sugar, white sugar, cinnamon, nutmeg, and vanilla extract, and give everything a stir.
Add the apples to the pan, and mix to thoroughly coat them with the spiced syrup. Allow the apples to gently simmer for 15 minutes, or until they are tender.
Serve warm with chopped nuts or ice cream, and store any leftovers in a sealed container in the fridge. Use within 3 days.