Preheat the oven to 375°F and cover a baking sheet with parchment paper.
Peel and core the Granny Smith apples, then cut them into approximately 1/2-inch pieces. Set the diced fruit aside.
Measure the water, sugar, cornstarch, cinnamon, and lemon juice before beginning the stovetop filling.
Add the water, cornstarch, sugar, ground cinnamon, and 1 1/2 teaspoons lemon juice to a medium saucepan. Place over medium-high heat and bring the mixture to a boil, stirring continuously for 3–4 minutes. Continue until the liquid becomes thick and glossy.
Stir in the diced apples, making sure the pieces are evenly covered with the thickened mixture. Reduce the heat to medium and simmer for 8–10 minutes, stirring from time to time.
Cook until the apples can be pierced with a fork while still retaining their shape. Remove the saucepan from the heat and allow the filling to cool slightly while preparing the cream cheese mixture.
Place the softened cream cheese in a large mixing bowl. Beat at medium speed for 2–3 minutes, or until completely creamy and smooth.
Add the 1/2 cup sugar and beat for an additional minute.
Mix in the egg yolk, 1 teaspoon lemon juice, and vanilla extract. Continue beating until the filling is light, uniform, and free of lumps. Set it aside while the cooked apples finish cooling.
Place the thawed puff pastry sheets on a clean work surface.
Divide the cream cheese mixture equally between the two pastry sheets. Spread it along the center of each sheet, keeping approximately 2 inches clear at both ends.
Spoon the cooled apple mixture evenly over the cream cheese layer.
At both ends of each pastry sheet, make two 1-inch cuts directed toward the filling without slicing into the filling itself. Fold these end sections inward to help enclose the filling.
Cut narrow diagonal strips along both long sides of each pastry at approximately a 45-degree angle. Alternately fold the strips over the center to form a braided pattern across the filling.
Whisk the large egg with 1 tablespoon water until evenly combined.
Using a pastry brush, apply a thin, uniform layer of egg wash over the exposed puff pastry. Avoid soaking the dough; a light coating is sufficient for an even golden finish.
Sprinkle coarse sugar over the pastries, including the braided sections.
Transfer the assembled pastries to the parchment-lined baking sheet.
Bake at 375°F for 25–30 minutes, or until the puff pastry has risen and developed a deep golden-brown color. If the oven heats unevenly, rotate the baking sheet halfway through the baking period.
Check that the undersides of the pastries are also golden and that the cream cheese filling is set while remaining slightly soft.
Allow the baked pastries to cool on the baking sheet for 5 minutes before serving.