Cut the dough into 9 (3×4-inch) rectangles—two cuts across, two cuts down. Carefully slide the parchment and dough back onto the baking sheet, making sure the rectangles are spaced about 1 inch apart. (Don’t worry if the space is a little tight, the dough won’t spread much while baking.)
Stir the jam filling to loosen it, then spoon about 1 tablespoon of jam onto the center of each rectangle. Use the back of a spoon to spread it into an even layer, leaving a 1/4-inch border around the edges. Pop the tray in the fridge while you roll out and cut the second piece of dough into another 9 rectangles.
In a small bowl, lightly beat 1 large egg with a fork. Remove the baking sheet from the fridge and brush the egg wash along the borders of each rectangle. Top with the remaining dough rectangles, pressing gently to seal. Crimp the edges with a fork and prick the tops of each pastry a few times to allow steam to escape. Refrigerate for another 30 minutes.
While the pastries chill, arrange a rack in the middle of your oven and preheat to 350ºF. Bake for 28 to 32 minutes, or until golden-brown all over. Let the pastries cool on the baking sheet for 5 minutes, then transfer them to a wire rack to cool completely, about 30 minutes. In the meantime, prepare the icing.