Preheat your oven to 400°F (200°C). Line a baking sheet with parchment paper or a silicone baking mat.

Place the thawed puff pastry sheet on a lightly floured surface. If the folds are still tight, let it sit at room temperature for about 5 minutes so it can unfold without cracking.

Gently roll the puff pastry into roughly a 10×12-inch rectangle, just enough to smooth and even it out. Transfer the pastry to the prepared baking sheet.

With a sharp knife, score a border about ½ inch from the edges all the way around the rectangle without cutting all the way through. Use a fork to poke holes all over the center area inside the border to help keep it flat while baking.

Lightly brush the scored border with the beaten egg and water mixture. You can lightly swipe the center, but focus most of the egg wash on the edges to encourage a golden, puffy crust.

Core the apples and slice them thinly, about ⅛–¼ inch thick. Peeling is optional; you can leave the peel on for color and texture or peel for a softer bite.

In a medium bowl, toss the apple slices with the melted butter, sugar, cinnamon, nutmeg (if using), and vanilla until the slices are evenly coated.

Lay the coated apple slices over the center of the puff pastry in slightly overlapping rows or a shingled pattern, staying inside the scored border. Avoid piling the apples too thickly so they bake evenly.

Bake on the center rack for 20–25 minutes, or until the edges of the puff pastry are deep golden brown and puffed and the apples are tender. If the edges brown too quickly, tent them loosely with foil during the last few minutes of baking.

While the tart bakes, combine the apricot jam or apple jelly with 1–2 teaspoons of water in a small saucepan or microwave-safe bowl. Warm until melted and smooth, stirring to combine. Add a tiny bit more water if needed to reach a brushable consistency.

As soon as the tart comes out of the oven, gently brush the warm jam mixture over the apples and, if desired, over the pastry edges to give the tart a glossy finish.

Let the tart cool on the pan for 10–15 minutes so the pastry can set and crisp. Slice into squares or rectangles and serve warm or at room temperature, plain or with ice cream or whipped cream.