Make the filling: Melt the butter in a large cast-iron skillet over medium-high heat. Add the the flour and whisk until bubbling.
Whisk in the milk until smooth, then add the chicken, peas and carrots, pearl onions, potato, salt, and pepper . Stir until everything is well coated, then continue to cook until the filling is bubbling and the liquid has thickened slightly, 7-10 minutes. Remove the pan from the heat.
Make the biscuits: In a large bowl, stir together the flour, baking powder, baking soda, and salt.
Using your a fork, your hands, or a pastry cutter, work the butter into the dry ingredients, leaving pea-sized chunks.
Add the buttermilk and fold with a rubber spatula until a thick dough comes together.
Turn the dough out onto a lightly floured surface. Bring the dough together into one large mass, then roll out with a rolling pin to about 1 ½-2 inches (4-5 centimeters) thick.
Using a 2-3-inch (5-7 centimeter) biscuit cutter or small glass dipped in flour, cut out rounds from the dough. Cut out all of the rounds, then place on top of the filling so the dough does not warm too much. Re-roll the dough scraps if needed. You should have about 10 biscuits total.
Brush tops of the biscuits with the beaten egg and top with cheddar cheese.
Bake for 25-30 minutes, until the biscuits are golden brown and cooked through. Carefully remove from the oven and let cool for 10 minutes. Garnish with chives before serving.