Melt the butter in a saucepan. Stir in flour to form a paste. Add water, white sugar and brown sugar, and bring to a boil. Reduce temperature to low and stir in cinnamon. Keep warm while you prepare the apples.
Grease an 8-inch round skillet, pan, pie plate (or equivalent sized) pan. Set on a baking sheet and set aside.
Peel and core apples, then thinly slice. Add to prepared baking pan, filling the pan about 2/3-3/4 full.
Using a sharp knife or pizza cutter, cut thawed puff pastry into irregular, 1-inch-ish shapes. Scatter about 1/3 of the pastry pieces randomly over top of apples. Slowly pour the sugar mixture over-top, pouring all around the pan to make sure the filling reaches all parts of the pan. Pour slowly so it drips down into the apples and not over the side. Place the remaining pastry pieces on top, overlapping, as necessary and leaving a few small open spots (to allow steam to escape). Sprinkle a bit of sugar on to the top of pastry.
Place into preheated oven (on baking sheet to catch overflowing juices!) and bake for 15 minutes, then reduce oven temperature to 350F and continue baking for an addition 35-40 minutes, or until filling is bubbly and pastry is golden brown and puffy. Let stand a few minutes before serving. Lovely served with a dusting of icing sugar and a scoop of ice cream.
Be sure to read the "Cook's Notes" in the original post, for more tips, options, substitutions and variations for this recipe!