Preheat oven to 325 degrees Fahrenheit.

Combine the flour, butter and brown sugar in a large mixing bowl and press between your fingers until small crumbs form. Add the pecan pieces and mix until evenly combined.

Core the apples three-quarters of the way down the center, leaving the bottoms of the apples intact.

Pour one tablespoon of maple syrup into each core and then fill with the crumb mixture.

Bake uncovered for 35 to 45 minutes, checking the apples often. Before serving, pour warmed caramel sauce over the apples.

Stir together the sugar, corn syrup and water in a heavy saucepan, until the sugar begins to dissolve.

Heat the mixture, stirring constantly until the sugar dissolves completely and the syrup is bubbling. At this point, stop stirring. Allow the mixture to boil undisturbed until it reaches 380 degrees Fahrenheit on a candy thermometer, or until its color is a deep amber.

Remove the pot from the heat and slowly and carefully add the cream, which will bubble furiously. Stir the mixture until it is smooth, scraping up the thick parts at the bottom. If any lumps develop, reheat the mixture until they have dissolved.

Add the butter and stir until combined.

Add the maple extract, if using.

Let the mixture cool for three minutes.

To soften before serving, either heat up the caramel in a bowl set on top of a pot of simmering water, or microwave it for one minute.