Preheat the oven to 400°F. Line a baking sheet with parchment paper and set aside.
In a large bowl, whisk together 2 cups all-purpose flour, 1 tablespoon baking powder, 1/2 teaspoon baking soda, and 1 teaspoon kosher salt.
Using a box grater, grate 3/4 cup cold unsalted butter into the flour mixture. Toss lightly to distribute the butter pieces, then transfer the bowl to the freezer for 5–7 minutes.
While the butter chills, whisk together 1/2 cup cold buttermilk (shaken) and 1 cold large egg in a measuring cup or small bowl until combined. In a separate small bowl, whisk together the remaining large egg and 1 tablespoon milk (or 1 tablespoon water) to make the egg wash. Refrigerate both mixtures until ready to use.
Remove the flour bowl from the freezer. Using your fingertips, gently break up any large clumps of butter and work the mixture until the butter pieces are the size of small peas and are evenly coated with flour.
Pour the buttermilk-and-egg mixture into the flour-and-butter mixture all at once. Stir with a fork or spatula just until the dough comes together and there are no large dry streaks of flour. Do not overmix.
Turn the dough out onto a lightly floured surface. Gently knead the dough 3–6 times until it holds together in a single mass. Pat or roll the dough to an even 3/4-inch thickness.
Using a 2 1/2-inch biscuit cutter, cut out biscuits by pressing straight down (do not twist). Gently transfer the cut biscuits to the prepared baking sheet. Gather scraps, re-pat to 3/4-inch thickness, and cut additional biscuits (you should end up with about 6–8 total).
If the butter has softened while you worked, place the baking sheet in the freezer for 5 minutes to firm the biscuits before baking.
Remove the baking sheet from the freezer and brush the tops of the biscuits with the egg wash. Bake at 400°F for 10–15 minutes, until the biscuits are tall and medium golden brown.
Remove from the oven and serve warm.