Preheat the oven to 375°F (190°C) and line a baking sheet with parchment paper.
In a large bowl, whisk together the all-purpose flour, cake flour, baking powder, sugar, and salt until evenly combined.
Using a pastry blender or fork, cut the cold, diced butter into the flour mixture until it resembles coarse crumbs with pea-sized pieces of butter remaining.
Toss the blueberries and chocolate chips with a little flour from the bowl (or add them directly) and gently fold them into the dry mixture to distribute evenly.
In a separate bowl, whisk together the buttermilk, two eggs, and vanilla extract until blended.
Pour the wet mixture over the dry ingredients and mix briefly with a fork until the dough just comes together; do not overmix.
Turn the dough onto a lightly floured surface and gently knead a few times until uniform. Divide the dough in half.
Shape each half into an 8-inch (20 cm) disc about 1 inch thick. Cut each disc into 8 wedges (like a pizza) for a total of 16 scones.
Arrange the scones on the prepared baking sheet. Brush the tops with the beaten egg for an egg wash and sprinkle with additional sugar if desired.
Bake for 20 to 25 minutes, until the scones are golden brown and cooked through. Remove from oven and cool slightly before serving.