Mix the dry ingredients. Whisk the flour, sugar, and salt in a large bowl.

Cut in the butter. Work in the cold butter with a pastry cutter or your fingertips until the mixture has coarse crumbs and some pea-sized pieces. Stop before the butter becomes soft or the mixture turns into a paste.

Add the discard. Stir in the cold discard. Add ice water 1 tablespoon at a time, using only enough for the dough to hold together when squeezed. It can look a little crumbly before you press it together; do not knead it smooth or automatically add all the water.

Chill the dough. Divide in half and press each half into a flat rectangle. Wrap and refrigerate for at least 1 hour, or up to 2 days.

Prepare the filling. Stir the cornstarch and cold water together in a small saucepan, then stir in the jam. Bring to a gentle bubble over medium-low heat, stirring, and cook about 1 minute until thickened. Transfer to a bowl and cool completely before filling the pastry. This is a pastry filling step, not a canning recipe.

Roll and cut. Line a baking sheet with parchment. On a lightly floured surface, roll one dough portion into a rectangle approximately 10 × 14 inches, about 1/8 inch thick. Trim or patch the edges as needed. Cut into 8 rectangles, each about 3 1/2 × 5 inches: split the 10-inch side in half and the 14-inch side into four. Repeat with the second portion so you have 16 matching rectangles. Keep the dough cold, returning it to the refrigerator if it softens.

Fill. Arrange 8 rectangles on the baking sheet, allowing space between them. Beat the egg and water together. Brush a little egg wash around the edges of each bottom rectangle. Place about 1 tablespoon cooled strawberry filling in the middle and spread lightly, keeping a 1/2-inch border clear on all sides.

Seal and vent. Top with the remaining rectangles. Gently press around the filling to remove trapped air, press the edges together, then crimp firmly with a fork. Poke 3–4 small steam holes in each top. Brush lightly with egg wash, avoiding heavy drips along the edges.

Chill again. Refrigerate the assembled pastries for 20 minutes while the oven heats to 375°F. If your tray is crowded, divide the pastries between two sheets and bake one sheet at a time, keeping the other refrigerated.

Bake. Bake for 25–30 minutes, until the pastry looks dry and set, with light golden edges and a lightly golden underside. Add a few minutes if needed; underbaked pastry can look pale and feel doughy. Let cool on the pan for 10 minutes, then transfer carefully to a rack and cool completely.

Ice and decorate. Whisk the powdered sugar, vanilla, and 1 1/2 tablespoons milk together. Add more milk a few drops at a time only if needed. The icing should spread slowly rather than run off the pastry. Spoon over the cooled pastries, leaving a little border. Add sprinkles before the icing sets.