Prepare Crust: Pulse almonds in a food processor 15 times or until finely ground. Stir together ground almonds and next 3 ingredients in a bowl. Cut cold butter and cold shortening into flour mixture with a pastry blender until mixture resembles small peas. Gradually stir in ice-cold water and vodka just until dry ingredients are moistened. (Add up to 2 Tbsp. ice-cold water, if necessary.) Shape into a disk. Wrap in plastic wrap; chill 3 to 24 hours.
Meanwhile, prepare Topping: Toss together 1 cup blueberries and next 5 ingredients. Let stand, stirring occasionally, 30 minutes. Cover and chill until ready to serve.
Unwrap chilled dough, and place on floured parchment paper; sprinkle with flour. Roll to a 14- to 15-inch circle. Carefully transfer dough and parchment paper to a jelly-roll pan.
Prepare Filling: Stir together 1/3 cup sugar and next 2 ingredients in a large bowl; stir in 2 cups each blueberries and blackberries. Spoon mixture into center of dough, leaving a 2 1/2-inch border around edges; dot with butter. (Do not make filling ahead or it will become too juicy.) Fold edges of dough around fruit, pleating as you go and sealing cracks. Brush dough with beaten egg, and sprinkle with 1 Tbsp. sugar. Freeze 15 minutes.
Preheat oven to 425°. Bake on lower oven rack 30 to 35 minutes or until crust is golden. Transfer to a wire rack; cool 30 minutes. Serve with Honey Sour Cream, if desired.