Whisk the flour, sugar, and salt together. Cut in the cold butter until the mixture resembles coarse meal with a few pea‑sized pieces.

Mix the egg with 2 tablespoons cold milk and stir into the flour mixture just until a dough forms. Press into a disc, wrap, and chill 30 minutes.

Roll the dough between parchment to ⅛‑inch thickness and cut into 8 rectangles. Stir the jam with the cornstarch. Place 4 rectangles on a parchment‑lined baking sheet, top each with 1 tablespoon of jam, leaving a border. Brush edges with water, add the top rectangles, and crimp edges with a fork to seal.

Preheat oven to 400°F (200°C). Bake for 16 to 18 minutes until golden. Cool completely on a wire rack.

Whisk the powdered sugar, 1 tablespoon milk, and vanilla; add the remaining milk only if needed. Ice the cooled tarts, top with sprinkles if using, and let set for 10 minutes before serving.