In a large mixing bowl, whisk together the flour, sugar, and salt. Add the butter and use your fingertips or a pastry cutter to work it into the flour mixture until pea-sized crumbs form.

Slowly drizzle in the cold milk, stirring until a soft dough forms. The dough should come together easily and feel smooth, but not sticky.

Transfer the dough to a lightly floured surface and shape into a thick disc. Wrap tightly in plastic wrap and refrigerate for at least 1 hour.

While the dough chills, prepare the compote. In a saucepan, combine the blueberries and sugar. Bring to a simmer over medium heat and cook for about 5 minutes, stirring occasionally.

In a small bowl, whisk together the cornstarch and 2 teaspoons water. Pour the slurry into the saucepan and stir constantly. The mixture will thicken quickly. Stir in the lemon zest, lemon juice, and vanilla. Continue simmering until thick and jam-like. Transfer to a small jar or bowl and refrigerate until completely cooled.

Remove the dough from the refrigerator and transfer to a lightly floured surface. Roll into a 12 x 16-inch rectangle. Trim any rough edges, then cut into a grid of 4 columns and 4 rows to create 16 equal rectangles.

In a small bowl, whisk together the egg and 1 tablespoon water to make an egg wash. Spoon about 1 tablespoon of compote onto the center of 8 rectangles, leaving a small border around the edges. Brush the borders with egg wash and top with the remaining rectangles. Seal the edges with a fork and poke a few holes in the tops for venting. Brush the tops with egg wash and refrigerate for 10 minutes.

Preheat the oven to 350°F. Bake for 20–25 minutes, or until golden brown. Let cool on the baking sheet for 10 minutes before transferring to a wire rack to cool completely.