Pulse flour, sugar, and salt in a food processor to combine.
Add cold cubed butter and pulse in short bursts until butter is reduced to pea-sized pieces with visible flecks.
Stir vinegar into ice water, then drizzle over flour while pulsing until dough just starts to clump together.
Turn shaggy dough onto a floured surface, divide in half, shape into disks, wrap, and chill for at least 30 minutes.
Combine diced strawberries, blueberries, and raspberries in a small saucepan.
Whisk sugar and cornstarch together, then sprinkle over berries and toss gently to coat.
Add lemon juice and salt, then cook over medium heat for 5 to 7 minutes until thickened. Cool completely before assembling.
Preheat oven to 400°F and line a baking sheet with parchment paper.
Roll one dough disk to 1/8 inch thickness and cut into 10 circles with a 3.5 to 4 inch cutter. Repeat with remaining disk.
Place half the dough circles on a work surface. Spoon 1 to 1.5 tablespoons of cooled filling into the center, leaving a 1/2 inch border.
Brush the exposed border with egg wash, then top with another dough circle and press edges to seal.
Crimp the sealed edges with a fork and transfer to the prepared baking sheet, spacing them 1 inch apart.
Cut 2 to 3 small slits in the top of each pie, brush with remaining egg wash, and sprinkle with turbinado sugar.
Bake for 22 to 25 minutes until golden brown and filling is bubbling through the vents. Cool on a wire rack before serving.