In a 10-inch cast-iron skillet, melt the 3 tablespoons butter over medium heat.
Add the carrots, celery, onion and rosemary, salt and pepper; cook, stirring occasionally, until tender; 10 to 12 minutes.
Add the flour and cook, stirring constantly, for 30 seconds, or until no white clumps of flour are visible.
Add the chicken stock, milk, 1-½ teaspoons salt and ½ teaspoon pepper; bring to a simmer over medium heat.
Turn the heat to low and simmer; stirring often and scraping the bottom of the pan, until very thick; 7 to 10 minutes; remove from heat.
Add the sherry, chives and chicken; stir until well combined.
Unwrap the chilled biscuit dough and place it on a well-floured surface.
Working quickly (you want the biscuits to be as cold as possible when they go into the oven), lightly press the dough with your hands into a rectangle about ¾-inch thick.
Fold the dough in half and press once more to a ¾-inch-thick rectangle.
Cut out biscuits with a 2½- to 3-inch-round cutter and arrange them on top of the filling.
Brush the tops of the biscuits with the egg wash.
Position the skillet on a sheet pan to catch any drips; place in the pre-heated oven and bake, uncovered, for 20 to 30 minutes or until biscuits are puffed and golden.
Remove from the oven and cool for 10 minutes to let the filling thicken slightly, then serve.