Whisk together the flour, baking powder and baking soda in a large bowl. Add the butter and use a pastry cutter, a fork or food processor to cut the butter into the flour until the pieces are about the size of peas.
Stir in the beaten egg with a wooden spoon. Next stir in the buttermilk. Stir just until it comes together (should be sticky still). Add a little more buttermilk or flour if needed. Cover the bowl and refrigerate 30-60 minutes.
Preheat the oven to 400°F and line a small baking sheet.
Place the dough onto a floured surface or silicone mat. Flour your hands and press down into a circle (about 1/2 inches thick). Use a floured 2 3/4" round cutter to cut 10 biscuits. Transfer to prepared baking sheet, so they are all touching.
Whisk together 1 egg and 1 Tablespoon of buttermilk. Brush onto the tops of the homemade biscuits (you want need to use it all). Bake until golden brown, 15-20 minutes.