Preheat the oven to 425 degrees F.
Place 2 cups of flour, baking powder, and salt in the bowl of a food processor or an electric mixer fitted with the paddle attachment. With the mixer on low, add the butter and Parmesan cheese and mix until the butter is the size of peas.
Combine the buttermilk and egg in a small measuring cup and beat lightly with a fork. With the mixer still on low, quickly add the buttermilk mixture to the flour mixture and mix only until moistened. Add the cheese to the dough. Mix only until roughly combined.
On a floured surface, roll the dough out to ¾-inch thick. With a circular biscuit cutter, cut dough out and transfer to a sheet pan lined with parchment paper. Brush the tops with the egg wash, sprinkle with shredded cheese and bake for 20 to 25 minutes, until the tops are browned and the biscuits are cooked through. Serve hot or warm.