Mix together the water and vinegar in a measuring cup and keep cold.
Combine the flour, sugar and salt in the bowl of a food processor and pulse a few times to combine. Scatter the cold butter over the top and pulse a few more times until the mixture resembles coarse meal. Add the water-vinegar mixture a few tablespoons at a time and pulse until the dough just comes together. Transfer the dough to a lightly-floured surface and knead gently until it just comes together. Form into a rectangle and cover with plastic wrap. Refrigerate for at least 1 hour and up to 48 hours.
While the dough is chilling, combine the fruit and sugar in a small saucepan. Cook over high heat until it comes to a boil, the fruit starts to break down and the juices start to release, about 8 minutes. Stir together the lemon juice and cornstarch until combined. Add the cornstarch mixture to the pot while it is still boiling and cook until it begins to thicken, about 1 minute. Stir in the jam and the lemon zest. Transfer to a bowl, and refrigerate until cold.
Preheat the oven to 375 degrees F. Line baking sheets with parchment paper.
Roll out half of the dough on a lightly-floured work surface into a large 1/8-inch-thick rectangle. Trim the sides with a knife so that you have a 9- by 8-inch rectangle with straight edges. Along the length, cut the rectangle into thirds so that you have three 3-inch strips. Then, cut the strips into fourths so that you have twelve small 3- by 2-inch rectangles in total. Carefully transfer these rectangles to one of your prepared baking sheets. Repeat this with the other half of dough. Gather the scraps of both halves and re-roll to create more rectangles (you will get 6 to 8 more).
Spoon a heaping teaspoon of strawberry filling in the center of half of the rectangles so that it is slightly heaped (do not spread it out). Make sure not to over-fill them and leave an about 1/4-inch border. Lay another rectangle over the filling and press gently around the edges to seal slightly. Use the tines of a fork to press the two layers of dough together and seal completely. Place the baking sheet with the pastries in the fridge until chilled, about 20 minutes.
Lightly brush the tops of the pastries with milk. Use a toothpick to poke 8 holes over the top surface to allow the steam escape during baking. Bake until golden around the edges, 15 to 20 minutes. Transfer to a wire rack to cool. Stir the milk into the confectioners' sugar until smooth (add a drop or 2 of water if you need to loosen it). Drizzle over the cooled pastries and serve.