Preheat the oven to 400°F/200°C.

After melting the butter in a medium-sized, deep skillet or pot, add the sliced apples and cook on medium heat for about 10 minutes, stirring occasionally. Reduce the heat to low and add the brown sugar, cinnamon, and cardamom. Continue cooking the mixture, stirring occasionally for about another 5 minutes. Let the mixture cool slightly, to make it easier to handle and so it won't melt the pastry dough.

Cut two parchment sheets, one as big as your sheet pan and the other slightly narrower. Lightly dust them with flour. Put the still-cool, defrosted pastry sheet on the larger parchment and roll it out to about 10 & 1/2 inches wide by 13 & 1/2 inches long. Cut the pastry almost in half lengthwise (I like to leave one side, for the top, slightly wider than the bottom, by about 1 inch/1 & 1/4 cm) and put the slightly larger/top half back back in the refrigerator on the narrower sheet of parchment, while you work with the bottom. Using a pizza cutter is the easiest way to cut the dough, but you can use a knife too.

Mound the apple slices horizontally (with their tips pointing to the narrower sides of the parchment) in a line down the center of the sheet of parchment, leaving a border around all the sides of about 1 inch/2 & 1/2 cm. Don't be too obsessive about how the apples look. (They will be covered by the top layer of puff pastry dough.) Just make an effort to keep them in a line facing the same way.

Brush the egg wash over the empty border. Take the second (top) sheet out of the refrigerator and gently place it on top of the dough with the apple slices lined up. Gently press around the edges to seal the pastry or seal it. Using a fork to seal works well, but you can do it with your fingers, rolling it slightly, or any other way that creates a tight seal.

Brush the top with the egg wash or vegan alternative, sprinkle the coarse sugar over the top and add a few vents with the sharp knife to let the air out of the pastry as it bakes.

Slide the parchment onto the pan and bake for 15-17 minutes, until the top of the puff pastry is golden. Cool to warm or room temperature before cutting.