Combine 2 1/2 cups flour and salt in a large bowl. Cut 1/2 cup butter and shortening into flour until the mixture resembles crumbs.

Mix egg, water, and white vinegar together in a separate bowl; pour in the crumb mixture and blend with a fork to form a dough ball. Wrap dough in plastic wrap and refrigerate until ready to use.

Melt 1/2 cup butter in a saucepan over medium heat. Blend 1/4 cup flour into the melted butter. Stir water and brown sugar into the butter mixture until smooth; remove from heat.

Roll chilled dough into a large rectangle on a flat surface dusted with flour; press into the bottom of an aluminum foil roasting pan and up the sides about an inch or more.

Dump the sliced apples onto the crust and gently spread into an even layer. Drizzle the butter mixture over the apples.

Pour apple crisp mix into a bowl. Cut 1/2 cup butter into the mix; scatter over the apples.