Preparation

1. 1.Add the flour, salt, and sugar to the work bowl fitted with the metal chopping blade and process for 10 seconds to sift. Add the butter and pulse until the mixture resembles coarse crumbs.
2. 2.Pour in water 1 tablespoon at a time, and pulse until mixture just forms a dough (you may not need all the water). Divide the dough into two even pieces, and form each into a square; wrap in plastic and refrigerate for at least 2 hours.
3. 3.Preheat oven to 300°F. Roll out one square of dough on a lightly floured surface to about a 13x9-inch rectangle. Transfer dough to a baking sheet lined with parchment paper.
4. 4.Cut 8 rectangles, each 4½ inches long and 3¼ inches wide. Chill until firm, 10 to 15 minutes. Repeat with the other piece of dough.
5. 5.Remove the first 8 rectangles of dough from the refrigerator and coat with about 1 tablespoon of the strawberry jam, leaving a ¼-inch border on each piece of dough. Transfer to the refrigerator to chill. Using a fork or skewer, poke holes in the remaining 8 rectangles.
6. 6.Remove the jam-coated pastry from the refrigerator, and brush egg yolk on the border of the pastry. Cover each jam rectangle with a rectangle with holes, aligning the two and pressing out any air pockets. Press around the borders to seal the two pastries. Chill for an additional 10 minutes.
7. 7.Bake until the bottoms are golden brown but not too dark on top, 35 to 40 minutes. Let cool.
8. 8.In a medium bowl, combine the sugar, half-and-half, and vanilla extract. When the pastries are cool, spread with the icing, and decorate with the sprinkles.