Roll 1 pie crust pastry onto a work surface; lightly roll with a rolling pin until crust is a 9x13-inch rectangle, cutting crust if needed. Press crust into the bottom of a 9x13-inch baking dish, placing leftover crust pieces in the corners.
Whisk white sugar, flour, cinnamon, and nutmeg together in a large bowl. Toss Granny Smith apples and gala apples in sugar mixture until evenly coated. Spread apple mixture evenly over pie crust; pour remaining juices over apples.
Roll the remaining pie crust to 9x13-inch rectangle and place over apple layer, folding edges over apples to completely cover.
Bake in the preheated oven until crust is lightly browned, 45 to 50 minutes. Cool apple bars slightly, about 20 minutes.
Whisk confectioners' sugar, milk, and vanilla extract together in a bowl until topping is smooth and desired consistency; drizzle over bars.