Pulse flour, sugar, and salt in food processor. Add shortening and butter; pulse until mixture resembles coarse crumbs.
Add water through feed tube, 1 tablespoon at a time, pulsing until mixture begins to hold together but still is crumbly. Shape pastry into a ball and flatten into a thick disk. Wrap and freeze 15 minutes. Meanwhile, arrange oven rack in lower third of oven. Heat oven to 425 degrees F.
Line a cookie sheet with parchment paper. Beat yolk and 1 tablespoon water in cup, and set aside.
Divide pastry into 4 equal pieces. Working with 1 piece at a time (refrigerate remaining 3 pieces) on a lightly floured surface, roll pastry into a 6-inch circle. Sprinkle 1 heaping tablespoon each dark and white chocolate in center of circle, leaving a 1-inch border. Brush border with egg and fold pastry in half. Seal edges with fingertips, then
trim with fluted pastry wheel. Transfer turnover with a spatula to cookie sheet, and brush top with egg mixture, then sprinkle with 1 teaspoon sugar. Refrigerate and repeat.
Bake turnovers until golden brown, 16 to 20 minutes. Serve