In a food processor, pulse the flour, 3 tbsp. sugar and the salt. Add the butter; pulse until the largest pieces are the size of a chickpea. Separate the egg, placing the yolk and white in separate bowls. Whisk 1/2 cup ice water into the yolk; add to the food processor. Pulse until moist clumps form, adding more ice water by the tablespoon if needed. Divide the dough in half. Form into two 4-by-7-inch rectangles; wrap each in plastic. Chill until firm, about 1 hour.
Preheat the oven to 400 degrees . On a lightly floured surface, roll out one piece of dough into a 12-by-17-inch rectangle; transfer to a jellyroll pan (1/2 by 10 by 15 inches), allowing the dough to slump into the corners. Trim the overhang to 1 inch; refrigerate. Roll out the remaining dough to a 12-by-17-inch rectangle. Cut lengthwise into 3/4-inch-wide strips.
In a bowl, whisk 1 cup sugar and the cornstarch. In a large bowl, drizzle the frozen berries with the vanilla, then mix in the cornstarch mixture. Scrape into the bottom crust, spreading in an even layer. Arrange 5 dough strips lengthwise over the fruit, then 8 strips crosswise, weaving if desired; trim the strips. Beat the egg white to loosen; using a pastry brush, brush the white on the edges of the bottom crust. Fold the overhang from the bottom crust over the strips; crimp decoratively. Brush egg white all over the dough. Sprinkle with the remaining 2 tbsp. sugar. Freeze the tart 30 minutes.
Bake until the crust is golden brown and the filling is bubbling, tenting with foil if browning too quickly, 45 to 55 minutes. Let the tart cool.