Melt butter in a large nonstick frying pan over medium heat. When the foaming subsides, add the vanilla, cinnamon, and salt. Sprinkle sugar evenly into pan and cook until bubbly.
Remove the pan from the heat and arrange the apples on top in a single layer. Return the pan to stove and cook over medium heat until the apples are glossy and knife tender, about 5 to 7 minutes. Remove the pan from the heat and add the rum. Carefully light the rum, with a long kitchen match. Return the pan to the heat and cook, shaking gently, until the flames subside. Transfer the apple mixture to serving dishes and serve immediately over ice cream.