1. Sandwich the softened butter between two sheets of parchment or waxed paper.
2. Flatten with a rolling pin as evenly as possible to form a roughly 4×5-inch rectangle refrigerate while you make the dough.
3. In a medium bowl, combine the flour and salt. Pour in the melted butter.
4. Gradually add water, and mix all the ingredients with your hands. Add more water, if necessary, to make a soft dough. Knead until you have smooth dough, about 1 minute.
5. On a lightly floured surface, roll the dough into an 8×10-inch rectangle
6. Remove the top layer of paper from the butter and invert the butter in the center of the dough so that the corners of the butter rectangle point toward the centers of the sides of the dough
7. Remove the other layer of paper. Fold the corners of the dough over the butter, like an envelope, covering it completely. Transfer to the baking sheet, cover, and refrigerate for 30 minutes
8. Dust the counter lightly with flour and lay down the butter and dough envelope, seam side up.
9. Lightly dust the top with flour and roll out to form a 12×15-inch rectangle. Don’t turn the dough as you do this. If any butter breaks through the pastry, just dab it with a little flour and carry on.
10. Lift one short end up and over the central portion of pastry, and then fold the other end over the central part, as if you were folding a letter.
11. Press the edges down lightly with a rolling pin to seal. Turn the pastry clockwise 90 degrees.
12. What was the top edge of your pastry will now be on the right. Mark the top left corner by pressing your finger into it. This is the first “turn.” The pastry and butter should still be firm enough to roll another turn, but if it is very soft and the butter is on the verge of oozing, put it on the baking sheet, cover, and chill for 20 minutes.
13. For your second turn, dust the work surface again with flour and put the pastry on it, with the marked corner at the top left. Dust with a little flour and roll the pastry out again, without turning the dough, into a 12×15-inch rectangle. Fold the pastry as before and turn it clockwise 90 degrees. Mark the top left corner with 2 indentations of the finger (this is to remind you that you have completed two turns). Transfer the pastry to the baking sheet, cover, and chill for 30 minutes.
14. Make another 4 turns, covering and chilling the dough for 20 minutes after each turn or every other turn, as needed. At the beginning of each turn place the pastry in front of you with the marked corner at the top left. After the third turn, make three indentations in the top left corner, and so on.
15. Wrap the pastry in plastic and chill for an hour (or overnight) before using. When to you are ready to bake let the pastry sit at room temperature until pliable before rolling, about 10 minutes.

Filling

1. In a large bowl, toss the plantain with the brown sugar, cinnamon and mix well.
2. In a 12-inch skillet, melt the butter over medium heat.
3. Carefully add the plantain mixture to the skillet; with a heatproof rubber spatula, scrape all the sugar and spices from the bowl into the skillet.
4. Stir the plantains to coat them with the butter and then spread them in a fairly even layer. Cook, stirring gently with the spatula every few minutes (try not to break the plantain slices), until the plantains are tender but not mushy (taste one) and still hold their shape about 3-5 minutes.
5. Scrape the plantain into a wide shallow dish or onto a baking sheet to cool completely before assembling the jalousie.

Assemble

1. Line a large rimmed baking sheet with parchment. In a small bowl, make an egg wash by beating the egg with 1 Tbs. water until well combined.
2. Unfold the puff pastry dough on a floured surface, and gently pinch together any seams that have split. With a floured rolling pin, roll the dough into an 11×14-inch rectangle.
3. With a sharp knife, cut the rectangle in half lengthwise to form two 5.5×14-inch rectangles.
4. Use a long spatula to help you move one of the dough rectangles onto the parchment-lined baking sheet.
5. Use a pastry brush to brush a 1-inch border of egg wash around the perimeter of the dough. (Save the remaining egg wash.) Arrange the fruit in a 4-inch-wide strip down the length of the dough. (You will get approximately 3 lines of fruit.)
6. Lightly dust the remaining piece of puff pastry with flour and then gently fold it in half lengthwise;
7. don’t crease the fold. Using a sharp knife, cut 1-1/2-inch-long slashes at 1-inch intervals along the folded side of the dough; leave at least a 1-inch border on the remaining three sides.
8. Do not unfold the dough. Using a long spatula, gently lift the folded strip and position it over the fruit-filled dough rectangle, matching up the straight edges.
9. Slash along the folded side.
10. Gently unfold the top piece of dough and stretch it over the filling, matching the straight edges all the way around the perimeter of the dough. Press the edges gently with your fingertips to seal the dough, and then, with a fork, very gently crimp the edges of the dough all the way around the pastry.
11. Unfold the dough over the filling.
12. Chill the assembled jalousie for 15 to 20 minutes. Meanwhile, position a rack in the lower third of the oven and heat the oven to 400°F.
13. Right before baking, brush the top of the jalousie with a very light coating of the remaining egg wash (you won’t need it all) and sprinkle with the demerara, turbinado, or granulated sugar.
14. Bake for 15 minutes and then rotate the baking sheet. Continue baking until the pastry is puffed, deep golden brown on top, and light golden brown on the bottom—use a spatula to gently lift the jalousie so you can peek underneath—another 10 to 15 minutes. Immediately transfer the jalousie from the baking sheet to a wire rack to cool for at least 45 minutes. (Easy way to remove hot jalousie is to lift the parchment to move the jalousie to the rack and then carefully slide the paper out from under the pastry.)