In a large bowl, whisk together the flour, sugar, and salt for the crust. Cut in the cold butter until the mixture resembles coarse crumbs.
Add ice water a tablespoon at a time until the dough comes together. Divide into two disks, wrap tightly, and refrigerate for at least 30 minutes.
Preheat the oven to 375°F (190°C). Place a baking sheet in the oven while it preheats.
In another bowl, gently toss the mulberries with granulated sugar, brown sugar, cornstarch, lemon juice, lemon zest, vanilla, and cinnamon.
Roll out one dough disk and place it into a 9-inch pie dish, leaving some overhang around the edges.
Pour the mulberry filling into the crust and dot the top with butter.
Roll out the second dough disk and place it over the filling. Trim excess dough, crimp the edges, and cut a few slits in the top for steam ventilation.
Whisk together the egg and milk, then brush the top crust evenly.
Place the pie on the hot baking sheet and bake for 45–50 minutes, or until the crust is golden brown and the filling is bubbling.
If the crust browns too quickly, loosely tent it with foil during the last 15 minutes of baking.
Remove the pie from the oven and let it cool on a wire rack for at least 30 minutes before slicing.