Preheat the oven to 350 degrees F. Butter a baking sheet.
Whisk the bread flour, cake flour, baking powder, sugar and salt in a large bowl. Work in the shortening with your fingers until the mixture is combined. Beat the eggs and buttermilk, then stir into the flour mixture with a fork to make a shaggy dough.
Turn the dough out onto a lightly floured surface and roll out to 1/4 inch thick. Cut out biscuits using a 2 1/2-inch-round cutter or glass, rerolling the scraps as needed. Transfer to the prepared baking sheet. Brush with the beaten egg.
Bake until golden and puffy, 18 to 20 minutes. Transfer to a rack and let cool slightly. Serve warm.