Preheat the oven to 375 degrees F. Line 2 baking sheets with parchment.
Whisk together the raspberry jam, 1 tablespoon of the lemon juice and 1 1/2 teaspoons of the cornstarch in a medium bowl until smooth. Fold in the raspberries. Whisk together the blueberry jam, remaining 1 tablespoon lemon juice and remaining 1 1/2 teaspoons cornstarch in another medium bowl until smooth. Fold in the blueberries. Set both fillings aside.
Unroll 2 pie crusts and place them on top of each other. Roll the dough out to a 14-by-15-inch rectangle. Place the dough over the top of a 12-cup muffin tin, making sure all of the holes are covered and some of the dough drapes over the sides. Roll out the remaining 2 pie crusts to a 14-by-15-inch rectangle and set aside (do not place over the muffin tin).
Using the muffin tin holes as a guide, spoon 1 tablespoon of raspberry filling in the center of 6 holes. Spoon 1 tablespoon of blueberry filling in the center of the remaining 6 holes. Whisk the egg with 1 tablespoon water in a small bowl. Brush the edges of the dough and the lines between the filling with the egg wash. Place the other rectangle of dough on top and gently press the edges to adhere the dough pieces together.
Place a cutting board on top of the dough on the muffin tin and flip it over so the dough is now on the cutting board. Trim the edges so you have 1 large rectangle with distinct corners. Use the tines of a fork to crimp the border. Cut the dough into 12 square pies--6 raspberry and 6 blueberry. Crimp the uncrimped sides.
Put the pies on the prepared baking sheets and brush the tops with the egg wash. Cut small slits in the top of each and sprinkle with the turbinado sugar. Bake until the pies are golden brown and some of the filling is bubbling through the slits, 15 to 17 minutes. Let cool completely on the baking sheets, about 30 minutes.
Use a sharp knife to make an incision in the cardboard of the ice cream pints from the top to the bottom. Slice each pint horizontally into four 1-inch discs and remove the cardboard. Reserve 2 of the ice cream discs for another use. Using the 6 remaining ice cream discs, sandwich 1 disc between 1 raspberry pie and 1 blueberry pie; repeat with the discs and pies.
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