Peel apples completely, removing all skin and stems. Halve apples across the middle creating both a top and bottom piece. Using a melon baller remove the core, being careful not to go completely through the bottom. Once the core is removed, continue using the melon baller to carve out the interior of the apple, keeping the small pieces of apple seperate.
In a small bowl, combine the carved out apple pieces with the white sugar, cinnamon, cornstarch, and softened butter. Mix until apple pieces are evenly coated.
Lay puff pastry out on a lightly floured surface. With a rolling pin, spread pastry out to make a sheet measuring roughly 9x14 inches (23x36cm). Using the round top of a sauce pan (ideally 6-7 inches (15-17 cm) in diameter), cut 2 circles of pastry out of each sheet and set them aside.
Preheat oven to 400˚F (200˚C).
Spoon filling into the hollowed-out core of the apples and place each filled apple in the middle of a round piece of pastry.
Apply a light wash of egg white to the rim of the pastry and fold toward the center over the apple, crimping the pastry as the edges meet. Continue until all the edges have been brought together. Use your hands to keep shape of the dumpling.
Once shaped, brush more egg white over the top of each dumpling. Lastly, using a small pairing knife, cut a small vent into the top of each dumpling.
Bake for 18-22 minutes in a preheated oven, or until pastry is golden brown and filling is bubbling.
Cider Sauce: While dumplings are baking combine water, apple cider, sugar, and cinnamon in a saucepan and bring it to a boil. Boil for 15 minutes allowing sauce to reduce and become thick.
Before serving, remove from heat and whisk in the 2 tablespoons of butter and lemon juice.
Serve by pouring sauce over dumplings while still hot.