Peel and core the apples and then slice them into 1/4-inch wedges. Melt the butter in a 10-inch non-stick skillet over medium heat.
When the butter has fully melted, sprinkle the brown sugar over the butter and then quickly place the apple slices in a single layer on top of the brown sugar.
With a wooden spoon, stir the apples to coat them in the butter and brown sugar. Continue to stir the apples.
When the butter and brown sugar have formed a syrup, sprinkle the cinnamon, nutmeg, and salt over the ingredients in the pan and stir to incorporate.
Reduce heat to medium-low and continue to stir the apples until they are fork tender and the syrup has reduced and thickened slightly.
Remove the skillet from the heat and let the apples cool for several minutes.
Serve alone, or as a topping for ice cream.