Preheat the oven to 425ºF (218ºC). Line a rimmed baking sheet with parchment; set aside.
Use the large holes of a box grater to grate butter (1 1/2 sticks) onto a plate. Place in the freezer until ready to add to the dough.
In a small bowl, toss the cheddar (1 cup) with 1/4 cup flour; set aside.
In the bowl of a stand-up mixer fitted with a paddle attachment, add 1 3/4 cup flour, baking powder (1 tablespoon), and 1 teaspoon salt.
Turn the mixer on low and add the butter. Mix just until combined.
Add the buttermilk (1/2 cup) and one egg to a liquid measuring cup and whisk with a fork.
Set mixer on low and quickly pour the buttermilk mixture into the flour mixture, mixing just until all the flour is moistened. Add the cheddar and flour mixture (be sure to add all the loose flour) to the mixer, while still mixing on low. Mix only until roughly combined.
Turn the dough out onto a well floured surface then knead quickly about 6 times.
Pat the dough out 1-inch thick. Cut with a 2 1/2-inch round cutter. Arrange biscuits on prepared baking sheet. Reroll the scraps to make one-two more biscuits. Place on baking sheet.
Beat egg with 2 tablespoons milk. Brush tops of biscuits with egg wash. Bake for 20 to 25 minutes, until the tops are browned and the biscuits are baked through.