Preheat oven to 425°F and place a rack in the center; line a rimmed baking sheet with parchment.
Whisk flour, baking powder, baking soda, salt, and pepper in a large bowl until combined.
Cut cold butter and shortening into flour with a pastry cutter or two forks until pea-size crumbs remain and mixture looks sandy with visible butter bits.
Toss chilled grated cheddar with 1 tablespoon flour, then stir into dry mix with chopped chives.
Pour cold buttermilk into dry ingredients and gently fold with a spatula until dough comes together but still shaggy; do not overmix.
Turn dough onto a lightly floured surface, fold over itself 2 times to create layers, pat to 1-inch thickness, and cut into eight rounds with a 2.5-inch cutter; place on prepared sheet with sides touching.
Brush tops with beaten egg or 2 tablespoons melted butter and bake at 425°F for 12-15 minutes until tops are golden brown and edges set.
Transfer to a wire rack and brush with remaining melted butter; serve warm.
