In a large bowl, whisk together the flour, baking powder, and baking soda.
Add the butter and use a pastry blender to cut the butter into the flour until the pieces are even and about the size of peas.
Stir in the beaten eggs with a wooden spoon until combined.
Stir in 1 ½ cups buttermilk until the dough comes together into a sticky mass. If it is too dry, add more buttermilk 1 tablespoon at a time.
Cover the bowl and refrigerate for at least 30 minutes and up to overnight.
Position a rack in the middle of the oven and preheat the oven to 400°F.
Line a baking sheet with parchment paper.
Scrape the dough onto a floured work surface. Use your floured hands to press it into a round roughly 14 inches across and about ½-inch thick.
Use a floured 2 ¾-inch round cutter to cut out about 20 biscuits.
Transfer the biscuits to the prepared baking sheet, arranging them so that they all are touching.
In a small dish, beat together the remaining egg and 1 tablespoon buttermilk. Brush the mixture on the top of the biscuits.
Bake until golden brown, 15 to 20 minutes. Let cool slightly in the pan on a rack.
Serve with strawberry jam or gravy, if desired.
