Preheat the oven to 400 degrees. Line a baking sheet with parchment paper.
Prepare the puff pastry by thawing one sheet then cutting it into 4 squares and place on the prepared sheet pan. Place into the refrigerator 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 1/4-inch-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 slices of apples diagonally across the pastry and placing two slices of apple on each side of the arranged slices.
Make the cinnamon sugar by combining the sugar and cinnamon in a small dish stir until mixed very well. Evenly sprinkle the apples with the cinnamon sugar and then dot them with the butter evenly.
Bake the tarts by placing them into the oven and baking for 30-40 minutes, until the pastry is browned and the edges of the apples start to brown. Remove from the oven and let cool for a few minutes before transferring them to a 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 the water together in a small saucepan over medium heat until bubbly. Brush the apples and pastry completely with the jelly mixture.