For the crust: In a large bowl, mix together the flour, granulated sugar and salt. Using a pastry cutter or two knives, cut the butter into the flour until it resembles tiny pebbles. Lightly beat the egg and add to the mixture. Next, add the cold water and vinegar. Stir until the dough just comes together but is still crumbly, using more water if necessary. Form the dough into a disc and wrap tightly in plastic wrap. Chill for 1 hour.
Preheat the oven to 400 degrees F. Line two sheet pans with parchment.
On a floured surface, roll half of the dough out into a large rectangle, 12 inches by 16 inches. Cut the long side into 4-by-4-inch wide strips. Cut each strip in half across the middle, making 8 equally sized rectangles (4-by-6-inch).
For the filling: Put 2 tablespoons of the jam into the center of four of the cut rectangles. With a pastry brush, brush egg wash around the edges. Place the remaining rectangle over the jam, crimping the edges with a fork to seal. Press lightly on the top to help distribute the jam evenly. Transfer to the prepared sheet pan. Repeat with the other half of the dough. Bake until barely golden, 16 to 18 minutes.
For the glaze: Meanwhile, sift the powdered sugar into a large bowl. Add the milk, jam and food coloring. Mix until smooth, then set aside.
Remove the tarts from the oven and allow to cool, 20 minutes. Spoon over the glaze and top with sprinkles. Let the glaze set for 1 hour before bagging up.