Pour the lemon juice over the lemons in a medium-sized pan. Sprinkle over the sugar, cornstarch, cinnamon, and salt.
Turn heat to medium and cook, stirring occasionally and gently with a rubber spatula until the apples are JUST BARELY softened.
Set them aside and let them cool slightly.
Preheat the oven to 400 degrees. Line a baking sheet with parchment paper.
Mix all ingredients for the crumble topping and set it aside.
On a lightly floured surface, roll out the puff pastry sheet to a 14x20-inch rectangle. Place it on the lined baking sheet.
Pour the apples into the center of the puff pastry sheet. Spread them out, leaving about 1-inch all the way around.
Sprinkle the crumble topping over the apples. Mix the egg and brush it over the edges of the puff pastry.
Transfer it to the oven and bake until the edges are deep golden brown. Mine took about 20 minutes.
Remove and serve!