In a large bowl, whisk together flour, cocoa powder, 3/4 cup sugar, baking powder, and salt.
Using a cheese grater, grate the butter into the flour mixture, and stir to distribute.
In a small bowl, whisk together heavy cream, vanilla, and egg.
Add the heavy cream mixture to the flour mixture, and stir with a wooden spoon until mixture just comes together - do not overmix!
Turn dough out onto a lightly floured surface, and knead about five times.
Pat dough into disk about 3/4-inch thick.
Using a 3-inch cookie cutter or biscuit cutter, cut as many rounds as possible.
Arrange two inches apart on a parchment paper or silicone mat-lined baking sheet.
Reform scraps of dough, and cut out as many rounds as possible. Arrange on baking sheet.
Bake at 425 degrees for 12-15 minutes, or until shortcakes appear set.
Remove shortcakes to a wire rack to cool completely.