Combine the flour, sugar, and salt in a large bowl or food processor. Use a pastry cutter or food processor blades to cut the butter and shortening into the flour mixture until they resemble coarse crumbs.

Sprinkle 2 tablespoons of the cold water over the flour mixture. Use a fork or the food process to mix the water into the flour, repeating with the remaining water, 2 tablespoons at a time, just until the dough starts to come together.

Divide the dough in two. Flatten into a disc and wrap tightly in plastic wrap. Refrigerate for 1 hour or freeze for 20 minutes, then allow the chilled dough to sit on the counter to soften slightly before rolling out.

Preheat oven to 425 degrees F. Roll out the bottom crust on a floured surface and use it to line a 9-inch pie plate.

Combine the raspberries, peaches, 3/4 cup of sugar, cornstarch, salt, lemon juice, and vanilla bean paste in a bowl. Toss to evenly distribute the cornstarch and sugar with the fruit.

Transfer the filling into the pie crust. If there is a lot of juice being released by the fruit, you can either discard or transfer to a microwave-safe bowl and heat for 30-60 seconds in the microwave to thicken before adding to the pie crust with the rest of the filling.

Roll out the top crust and slice into strips of varying width using a pizza cutter or sharp knife. Lay the strips across the filling in one direction with a little space between them. Alternate lifting back every other strip and laying strips across going the other direction. Then replace the folded back strips and fold back the other strips, laying another strip across in the opposite direction. Repeat this process to make a lattice crust, then trim and crimp the edges with the tines of a fork.

Beat the egg in a bowl to make an egg wash, then brush on top of the lattice crust. Sprinkle with the remaining tablespoon of sugar.

Cool completely for 4-6 hours before slicing and serving.