Prepare dough: In a food processor, combine flour, salt, and sugar; pulse to combine. Add butter; pulse until mixture resembles coarse meal, with just a few pea-size pieces of butter remaining. Sprinkle with 4 tablespoons ice water. Pulse until dough is crumbly but holds together when squeezed with fingers (if necessary, add up to 4 tablespoons more ice water, 1 tablespoon at a time). Do not overprocess.
Divide dough into two equal portions; form each into a square 3/4 inch thick. Wrap tightly in plastic, and refrigerate until firm, at least 1 hour and up to 3 days.
Prepare filling: In a medium bowl, toss together berries, sugar, cornstarch, and lemon juice; set aside.
Preheat oven to 425 degrees. On a lightly floured work surface, roll out each square of dough to an 8-by-12-inch rectangle; cut each into six 4-inch squares (you should have 12 squares total). Dividing evenly, spoon filling onto center of each square. Brush two adjoining edges of each square with some egg wash. Fold these edges over filling to form a triangle; press firmly to seal (dough should be tightly pressed around filling). Crimp edges.
Transfer turnovers to a parchment-paper-lined baking sheet. Brush tops with remaining egg wash; sprinkle with sugar. With scissors, snip centers to create a steam vent. Bake until golden brown and crisp, 20 to 25 minutes, rotating sheet halfway through. Serve warm or at room temperature.
You can substitute peaches (cut into chunks) for the berries, or make a combo with equal amounts of both. Enjoy these turnovers the day they're made.