In a large bowl, add 2 1/2 cups of the flour, the granulated sugar, and the salt and whisk to combine.
Use a pastry cutter or two forks to cut the butter into the flour mixture until it resembles pea-sized bits.
Drizzle 1/2 cup of the ice water, 1 tablespoon at a time, over the flour mixture and use a spatula to stir after each addition until the dough begins to form large clumps. Note: If the dough is too dry, add the extra ice water, but be careful not to add more than needed.
Use the extra flour to dust a work surface.
Transfer the dough to the prepared work surface and use the extra flour to dust your hands. Use your floured hands to gently bring the dough mixture together into a ball. Avoid overworking it. If it feels too dry or crumbly, dip your fingers into the extra ice water. If it feels too sticky, sprinkle more of the extra flour over the dough.
Use a sharp knife or bench scraper to cut the dough ball in half.
Gently flatten each of the dough halves into 1-inch-thick discs, then tightly wrap them with plastic wrap.
Transfer the wrapped dough discs to the refrigerator and chill for at least 2 hours and up to 5 days.
Line two baking sheets with parchment paper.
Transfer 1 of the dough discs from the refrigerator and allow it to sit at room temperature and let it rest, about 5 minutes. Keep the remaining dough disc in the refrigerator.
Use the extra flour to lightly dust a work surface.
Unwrap the dough disc and place it on the prepared work surface. Use a rolling pin to roll it into a 9x12-inch rectangle that is 1/8-inch-thick. Trim the sides, as needed.
Use a pizza cutter or a sharp knife to horizontally cut the dough rectangle into 3 evenly-sized portions. Then, repeat vertically to create 9 (3x4-inch) rectangles.
Evenly place the dough rectangles onto one of the prepared baking sheets, then transfer it in the refrigerator to chill.
Transfer the remaining dough disc from the refrigerator and repeat the cutting process from steps 10-14.
While the dough rectangles are chilling, in a small bowl, add the brown sugar, 2 teaspoons of the cinnamon, and the remaining 1 tablespoon of the flour and stir to combine.
In a small bowl, add the egg and 2 teaspoons of the milk and whisk to combine.
Transfer one of the baking sheets from the refrigerator and evenly brush 1/3 of the egg wash over the entire surface of each of the dough rectangles. These will be the bottoms of the pastries.
Place 1 heaping tablespoon of the cinnamon filling in the center of each of the bottom dough rectangles, then use the back of a spoon to evenly smooth it out over the surface, leaving a 1/4-inch-thick border around the edges.
Transfer the remaining baking sheet from the refrigerator, then brush 1/2 of the remaining egg wash over the surface of the dough rectangles. These will be the tops of the pastries.
Place 1 of the top pastries, egg-wash-side up, on top of the filling-topped bottom pastries and use your fingertips to firmly press around the pockets of the filling, sealing the dough well on all sides.
Use a toothpick to poke 6-8 holes in the tops of each of the filled pastries to allow steam to escape.
Use a fork to crimp the edges of each of the pastries, tightly sealing them.
Transfer the baking sheets to the refrigerator and chill the filled pastries for at least 20 minutes and up to 1 hour.
Preheat the oven to 350 degrees F.
Transfer the baking sheets from the refrigerator and evenly brush the filled pastries with the remaining egg wash.
Transfer the baking sheets to the oven and bake the filled pastries, rotating the sheets halfway through, until they are golden-brown, about 25-30 minutes.
Transfer the baking sheets to a wire rack and let them rest until they are completely cooled.
In a medium bowl, add the confectioners' sugar, the remaining 1 tablespoon of the milk, the remaining cinnamon, and the vanilla and whisk until it is thick but spreadable. If it is too thick, add the extra milk, 1 teaspoon at a time, until it is spreadable.
Spoon the cinnamon glaze over the tops of each of the cooled pastries and use the back of a spoon to evenly spread it out.
Let the glazed pastries rest until the glaze is set, about 1 hour.
Serve.