Preheat oven to 325 degrees. Butter and line a 9-inch pan with parchment paper. Let the sides of the paper extend over the edges so you can lift the bars out of the pan.
Place butter in a medium microwave-safe bowl and microwave for 20 second increments until melted. Let cool slightly.
In a large bowl, whisk together the eggs, sugar, and vanilla for a good 1-2 minutes. Whisk in melted butter.
In a medium bowl, combine flour, baking powder, and salt.
Fold dry ingredients into wet just until incorporated. Fold in apples and butterscotch morsels.
Pour batter into prepared pan and bake for about 40 minutes. They should still be a little gooey in the middle. Let cool before cutting into squares.
For the caramel sauce, place brown sugar, half and half, and butter in a small saucepan. Heat over medium-low heat, whisking gently for 6 to 8 minutes, or until gets thicker. Remove from heat and add in vanilla extract and salt. Cool slightly before using.
To serve, top an apple bar with ice cream, drizzle with caramel sauce and add a few apple chips on top if desired.