Make Piecrust. Whisk flour and salt together in a large bowl. Cut in butter and shortening with a pastry blender until mixture resembles coarse meal. Sprinkle ice water, 1 tablespoon at a time, over mixture, incorporating with a fork until pastry is moist enough to hold together. Divide in half; shape each half into a disk and wrap in plastic. Refrigerate for at least 2 hours or overnight.
Heat oven to 400 degrees F. Roll one crust out to a 12-inch circle. Gently transfer to a 9-inch pie plate. Prepare Filling. In medium-size saucepan, combine 3/4 cup of the sugar, the tapioca, cornstarch and salt. Whisk to blend. Add juice and stir until smooth. Cook, stirring, for 5 minutes, until thickened and bubbly. Remove from heat and stir in cherries and raspberries.
Roll out remaining crust to a 12-inch circle. With a pastry cutter or pizza wheel, cut into 1-inch strips. Transfer pie filling to pastry-lined pie plate. Weave pastry into a lattice, alternating strips over and under each other. Fold edges under; crimp together. Brush pastry with egg yolk-water mixture, then sprinkle with remaining 2 teaspoons sugar.
Bake pie at 400 degrees F for 15 minutes. Reduce heat to 375 degrees F and bake an additional 45 minutes, covering pie edge with foil if browning too quickly. Cool at least an hour before slicing.
You may substitute refrigerated rolled piecrusts, if desired.