This mouthwatering apple dessert is from "Demolition Desserts" by Elizabeth Falkner.
Combine 1/2 cup sugar, 2 tablespoons water, and lemon juice in a medium skillet over high heat. Continue heating mixture, without stirring, until sugar has dissolved. Slowly tilt the skillet back and forth until sugar is caramelized and is the color of dark amber, 4 to 5 minutes.
Remove skillet from heat and carefully add apple cubes, arranging them in a single layer. Place the skillet over medium heat and allow the apples to simmer in the caramel until they slowly begin to darken and absorb the caramel, about 5 minutes.
Remove skillet from heat and sprinkle apples with vanilla and salt. Let stand, without stirring or moving apples, until all the caramel has been absorbed, 10 to 15 minutes.
Meanwhile, in a small saucepan over medium heat, combine apple juice, vinegar, and remaining 2 tablespoons sugar. Bring mixture just to a boil and immediately reduce heat to medium-low. Continue cooking until liquid has reduced by half, about 20 minutes.
To serve, divide apples evenly between 4 plates. Place a small scoop of ice cream next to the apples. Sprinkle apples and ice cream with cheddar cheese crumbles. Drizzle apple-balsamic reduction onto the plates or directly over the ice cream. Serve immediately. Reprinted with permission from Elizabeth Falkner's "Demolition Desserts: Recipes from Citizen Cake" by Elizabeth Falkner. Copyright 2007. Published by Ten Speed Press.