Rinse the apples, then peel, quarter, and core them with a sharp knife. Cut into thin slices, then toss with lemon juice so they don't turn brown.
Melt the butter in a large skillet over medium-high heat until bubbly. Stir in the slices to coat with butter, and cook until the juices have thickened slightly and they are beginning to soften, about 3 minutes.
Add the sugars, cinnamon, nutmeg, and salt and stir well. Turn the heat down to medium and cook until the slices are fork tender. Be careful not to overcook.
If the juices haven't thickened to your liking, remove the fried apple slices with a slotted spoon and whisk the cornstarch with a little water. Add the slurry to the sauce and simmer until thickened. Gently stir the slices into the thickened sauce.
Taste and adjust the spices as desired. Serve warm with whipped cream or ice cream for dessert, or with pork chops for a sweet and savory twist.
Crock Pot Method: Combine all the ingredients, including the cornstarch, in a slow cooker and cook on high for 1 ½ hours, or until the apples are tender. Check after 1 hour.