Preheat oven to 425 F. Lightly spray muffin tin cups (or mini tart pans) with cooking spray.
Make the Apple Filling
Melt butter in a heavy bottom saucepan over medium-high heat.
Add flour and stir quickly until combined.
Add 3 tablespoons water and sugar and mix well. Cook for approximately 5 minutes until it becomes a caramel sauce.
Remove from heat and allow it to cool.
Meanwhile, dice the apple and toss with ground cinnamon.
Combine caramel sauce and apples, mix well.
Prepare the Pies
Cut 12 circles from the pie dough using a 3.5-inch round biscuit cutter or cookie cutter.
Tuck the pie dough circles snugly into the cups.
Divide the apple filling evenly among the cups. Fill them full, but flush, not mounded.
Cut the rest of the pie dough into thin strips. Layer the strips on top of the mini pies by weaving over and under and alternating each one.
In a small bowl, whisk the egg and a teaspoon of water until frothy.
Brush the tops with the egg wash. If you would like to add handles, please see "notes" below.
Bake for 20-25 minutes, until light golden brown and the apple filling is bubbly.
Allow pies to cool in pan on a cooling rack for 10-15 minutes. Remove from tart pans or muffin tin and cool completely on wire rack.
Notes
To add a cute basket handle, cut strips of pie dough and twist. Bend those twists into a curve, the opening between the ends should fit over the diameter of the pies. Bake these following the instructions above, they may need to come out a little sooner, so keep on eye on them. Carefully insert toothpicks into each open end and poke the other end of the picks into the mini apple pie.
You can add a dusting of powdered sugar for decoration if you like.