01 - Combine flour, sugar, and salt in a large bowl. Add cold butter and blend with pastry cutter or fingertips until mixture resembles pea-sized crumbs.
02 - Incorporate sourdough discard and 2 tablespoons cold milk or water. Mix gently until dough begins to form, adding extra liquid if necessary.
03 - Divide dough into two portions, shape into disks, wrap tightly in plastic, and refrigerate for at least 1 hour.
04 - Set oven to 375°F and line a baking sheet with parchment paper.
05 - Roll each chilled dough disk on a lightly floured surface to 1/8-inch thick. Trim into sixteen 3x4-inch rectangles.
06 - Arrange 8 rectangles on the baking sheet. Place 1 tablespoon jam in center of each rectangle, leaving a 1/2-inch border.
07 - Brush edges with egg wash. Cover with remaining dough rectangles and seal perimeters by pressing with a fork.
08 - Poke several holes in the tops using a fork for steam release. Apply a coat of egg wash to surface.
09 - Place in oven and bake for 22 to 25 minutes, or until pastries are golden brown. Transfer to wire rack for cooling.
10 - Whisk together powdered sugar, milk, and vanilla extract until smooth. Drizzle over cooled pastries and decorate with sprinkles if desired.