In food processor, pulse flour, 1½ tbsp sugar, and salt until combined. Add the butter and pulse until fine crumbs form. Drizzle in ½ cup cold water in 2 batches, pulsing between, until dough holds together when squeezed. Transfer to large bowl and gently knead just until dough holds together. Divide into 4 mounds and shape into flat rectangles. Wrap each tightly in plastic. Refrigerate just until firm, about 35 min.
Meanwhile, preheat oven to 400°F. Juice and zest lemon into another large bowl. Add blueberries and blackberries, cornstarch, and remaining 4 tbsp sugar. Fold to combine. Hull and thinly slice the strawberries.
On floured work surface, roll 1 piece of dough into 12×9-inch rectangle. Transfer to 9×13-inch rimmed baking sheet, placing 9-inch side of dough along longer side of pan. Repeat rolling with another piece of dough. Transfer to other side of baking sheet, overlapping slightly with first piece of dough. Press seam together tightly to seal. Trim and shape edges of crust. Repeat rolling with remaining 2 pieces dough. Use cookie cutter to cut out stars from dough.
With slotted spoon, transfer berry mixture to crust in an even layer, leaving excess liquid in bowl. Arrange strawberries on top. Place cutouts over filling. In small bowl, beat the egg and 1 tbsp water. Brush egg wash onto cutouts and edges of crust. Bake 45–50 min., until filling is bubbling and crust is deep golden brown. Let cool completely before cutting.