Preheat the oven to 425°F. Line a baking sheet with parchment paper.
Whisk one-half cup of buttermilk together with one cold egg in a small bowl. Set aside.
In a large bowl, combine two cups of Bisquick and three-quarters of a cup of cold, cubed butter. Using a pastry cutter, blend until the mixture becomes coarse and crumbly. Mix in one cup of cheddar cheese. Add the buttermilk mixture and mix everything together. Use your hands to knead a few times until the dough barely comes together.
Turn the dough onto a Bisquick floured surface and continue kneading a few times. Add more Bisquick if needed.
Roll out the dough to a ¾-inch thickness.
Using a biscuit cutter or a glass, cut into biscuits.
Place the biscuits onto the prepared baking sheet. Repeat until you've used up all the dough.
Brush the top of each biscuit with the beaten egg and bake in the preheated oven for about twenty minutes. Transfer to a rack to cool.