Preheat the oven to 400°F (204°C). Line 2 sheet pans with parchment paper and set aside.

Gently roll both sheets of puff pastry just to seal the seams. Using a pizza cutter or sharp paring knife, cut each puff pastry square in half length-wise. Then, cut each strip of puff pastry into 3 equal rectangles width-wise so you have 12 puff pastry rectangles total.

Use a paring knife to score a 1/4-inch border around the edges of the puff pastry rectangles. But don't cut all the way through! Then, use the tines of a fork to dock the puff pastry in the centers, avoiding the borders. Place the puff pastry rectangles on the prepared sheet pans about 2-inches apart. (They will puff up quite a bit.) Refrigerate the pastry while you make the filling.

Peel and core the apples. Then, cut the apples into thin slices, about 1/8-inch thick. Spoon the apples into a large mixing bowl.

Add the dark brown sugar, melted butter, cornstarch, apple pie spice, cinnamon, and a pinch of salt. Toss until the apples are well-coated and the cornstarch is dissolved.

Next, whisk the egg with 1 tablespoon (15 milliliters) of water until smooth. Use a pastry brush to brush the puff pastry rectangles with egg wash.

Layer slices of apples down the center of each puff pastry rectangle in a shingled pattern. Don't be afraid to pack the apples in there until you're all out of slices. If you have any leftover brown sugar-spice mixture in the bottom of the bowl, spoon it over the apples.

Bake the tartlets for 18-20 minutes, until the apples are tender and the pastry is a deep golden-brown. Let the tartlets cool for a few minutes and then drizzle them with caramel sauce. Serve with ice cream or a dollop of whipped cream and enjoy!