Preheat oven to 400°F. Remove frozen puff pastry from the freezer and thaw at room temperature for 45 minutes, or until you can roll it out without it breaking.
Meanwhile, cook the apples. Heat the oil in a large frying pan over medium heat and add the apple wedges. Cook for five minutes, stirring often. Add the granulated sugar, cinnamon and ginger and cook for 20 minutes, or until fork tender, stirring often. Turn down the heat to low after ten minutes. If some pieces soften sooner than others, remove them to a large bowl. When cooked, place the apples into the bowl and let cool. If rushed, place the bowl of apples in the freezer for 15 minutes.
When the pastry is ready to be rolled, place a piece of parchment on the counter and sprinkle with flour. Unroll one of the pastry sheets and place on the parchment with the creases vertical. Roll with a rolling pin to smooth out the creases and to roll out the rectangle about 2 ½ inches larger on each side, trying to keep the rectangular shape. Every once in a while, lift up the pastry and sprinkle a little flour underneath.
Take a 9 X 13-inch pan and turn onto the pastry so the pan bottom is facing up. Holding the parchment and pastry to the pan rim, turn the pan over so the pastry is now in the bottom of the pan. Press into the corners and sides and remove the parchment. Use a fork to prick some holes in the pastry and place in the freezer until ready to bake.
Sprinkle some flour on the parchment you just used and roll out the second sheet of pastry about 14 inches wide and 16 inches long. Use a knife to cut the long way into 1-inch long strips of dough. Cover a cookie sheet with more parchment and lift up the strips and place them on the covered cookie sheet, about ½ inch apart. Place into the fridge.
When the apples are cooled, add the vanilla and honey and stir. Add the flour and brown sugar and stir to distribute; the apples with be coated with a gooey glaze. Scoop into the chilled pastry-lined 9 X 13 inch pan and spread evenly.
To assemble the lattice: Place half the dough strips (about 7) diagonally across the pie, about 1/2 inch apart. Fold the first, third, fifth and seventh strips back over themselves. Place another strip vertically across the unfolded strips. Unfold the folded strips over the new strip. Now fold back the strips that were not folded before, the second, fourth and sixth strips. Add another vertical strip, about ½ inch from the other, over the unfolded strips and then unfold the folded ones. Keep alternating until you have covered the pie. Trim off the edges and press to seal the edges of the lattice strips to the rim of bottom crust.
Use a pastry brush to brush the egg white all over the top of the lattice. Sprinkle the remaining teaspoon sugar all over the top. Bake for 35-40 minutes, or until pastry is golden. Serve warm. May be made a day in advance and stored covered at room temperature and reheated.