Whisk 250g flour, granulated sugar, and salt. Grate the cold butter into the bowl and toss to coat. Keep visible butter pieces; chill if they soften.
Add the cold discard and vinegar. Mix gently, then press together by hand. If a handful won't hold together and loose flour remains, add ice water 1 teaspoon at a time, checking after each addition. Do not knead smooth.
Divide into two equal portions and flatten into rectangles. Wrap and refrigerate at least 1 hour or up to 48 hours.
Mix brown sugar, cinnamon, and the 10g flour. Beat the egg with 1 teaspoon water separately.
Roll one dough portion on a lightly floured surface large enough to trim to 10 by 12 inches, approximately 1/8 inch thick. Cut the 10-inch side in half and the 12-inch side into four sections to make eight 3-by-5-inch rectangles. Transfer to parchment-lined sheets, 1 inch apart. Refrigerate while rolling and cutting the second portion into eight matching tops.
Brush the bases lightly with egg wash. Divide the filling among them, about 14g or roughly 1 tablespoon each. Spread thinly, leaving a clean 1/2-inch border.
Add the tops, gently press out trapped air, and press the borders together. Crimp with a fork, then poke each top in three or four places. Refrigerate 20 minutes. Heat the oven to 375°F (190°C), conventional setting.
Brush chilled tops lightly with egg wash. Bake 25–30 minutes, checking at 22 minutes, until golden on top and lightly browned underneath. If using two racks, rotate and swap pans after about 18 minutes.
Rest on the pans 10 minutes, then transfer to a rack to cool completely, about another 30–45 minutes.
Stir powdered sugar, cinnamon, vanilla, and 2 teaspoons milk until smooth. Add more milk a few drops at a time, up to about 3 teaspoons total, for a spreadable icing. Ice the cooled pastries. Allow an additional 30–60 minutes to firm before stacking if needed.