In a large bowl, whisk together flour, baking powder, baking soda, and salt.
Add the butter pieces and using either a pastry cutter or your hands (I like to use my fingers), cut (rub and flatten) each piece of butter into the flour mixture until small pea-sized pieces remain.
Mix together buttermilk and eggs in a measuring cup and pour this into the flour mixture. Stir with a spoon until a sticky, wet dough forms.
Cover the bowl and place it in the refrigerator to chill for at least 30 minutes, up to overnight.
Remove the dough from the refrigerator. On a lightly floured surface, knead dough briefly just to bring it together. Roll or pat out the dough to about 1 inch thick. Cut with a biscuit cutter.
Place the biscuits side by side (touching) in a ¼ sheet pan (9 by 13-inch) or similar size baking pan. Chill while preheating the oven to 400° F.
Brush the biscuits with melted butter.
Bake for 30-35 minutes, or until lightly browned and very well risen.
Cool for about 5-10 minutes before serving.
