Preheat the oven to 400 degrees. Line a baking sheet with parchment paper.
Thaw the puff pastry sheet at room temperature.
Cut it into 4 squares and place them on the prepared sheet pan. Refrigerate while you prepare the apples.
Prepare the apples by peeling them and cutting them in half through the stem. Remove the stems and cores with a sharp knife. Slice the apples crosswise into ¼inch-thick slices. Side Note: Cover the peeled & sliced apples with a wet paper towel while you finish peeling & slicing the other apple, to help prevent browning.
Prepare the tarts by placing overlapping apple slices diagonally across the pastry.
Make the cinnamon sugar by combining the sugar and cinnamon in a small dish.
Sprinkle the apples with the cinnamon sugar, and then dot them with butter.
Bake the tarts for 30-40 minutes, until the pastry is browned and the apple edges start to brown.
Remove from the oven and let cool for a few minutes before transferring them to a wire cooling rack. Side Note: The apple juices will burn on the baking sheet, but the tarts will be fine.
Make the glaze by heating the apricot jelly and water in a small saucepan over medium heat until bubbly. Brush the apples and pastry completely with the jelly mixture.