Place 2 cups of flour, the baking powder, and salt in the bowl of an electric mixer fitted with the paddle attachment. With the mixer on low, add the butter 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. In a small bowl, mix the Cheddar with small handful of flour and, with the mixer still on low, add the cheese to the dough. Mix only until roughly combined.
Dump out onto a well-floured board and knead lightly about 6 times. Roll the dough out to a rectangle 10 by 5 inches. With a sharp, floured knife, cut the dough lengthwise in half and then across in quarters, making 8 rough rectangles. Transfer to a sheet pan lined with parchment paper. Brush the tops with the egg wash, sprinkle with salt, if using, and bake for 20 to 25 minutes, until the tops are browned and the biscuits are cooked through. Serve hot or warm.