Heat the oven to 400°F (200°C) and line a baking sheet with parchment paper. Combine the apples, brown sugar, cinnamon, nutmeg, flour, and lemon juice in a bowl, then let the mixture stand for 5–10 minutes.

Unfold the cold puff pastry on a lightly floured surface and gently roll it into a rectangle if necessary. Place the apple mixture in a strip down the center third, leaving room at the top and bottom.

Cut diagonal strips about 1 inch wide on both sides of the pastry. Fold the strips alternately over the apples to create the braid, then pinch or tuck the ends closed.

Beat the egg with the water and brush the egg wash over the exposed pastry. If the assembled braid feels soft, refrigerate it for 10–15 minutes before baking.

Bake for 22–28 minutes, until puffed and deep golden brown. Cool the braid for at least 10 minutes before drizzling it with glaze.

Whisk the confectioners’ sugar, milk or cream, and optional vanilla into a smooth glaze. Drizzle it over the slightly warm or cooled braid and slice to serve.