Mix flour, sugar, and salt in a large bowl. Work in the butter with a pastry blender or fork until it’s broken into pea-sized pieces.
Add the yogurt and water and stir just until it comes together in a ball (do not knead). Wrap the ball in plastic wrap and flatten into a disc. Refrigerate for at least 1 hour.
Toss the fruit with sugar, lemon juice, vanilla, salt, and starch in a bowl.
Heat the oven to 400°F.
Place the disc of dough between two (16-inch) sheets of plastic wrap. Roll out the pastry between the sheets of plastic wrap until it’s a roughly 14-inch circle. Gently peel off the top layer of plastic wrap. Flip the pastry upside-down onto a parchment-lined baking sheet. Peel off the other layer of plastic wrap.
Pour the fruit and juices into the center of the pastry, leaving 2–3 inches around the edges. Lift the edges of pastry up toward the fruit, folding between each section.
Whip the egg with 1 tablespoon of water and brush over the exposed pastry, then sprinkle turbinado.
Bake for about 30 minutes, or until the crust is deeply golden brown. Allow to cool for at least 10 minutes before slicing. Top with ice cream or whipped cream, and garnish with mint leaves if you’re feeling fancy.