(READ RECIPE AND NOTES BEFORE PREPARING!) Preheat oven to 400 degrees. Grease a 7x11 or 8x12 baking dish with non-stick spray or butter.
Heat a large skillet to a medium heat. Add 3 tablespoons of butter. When butter melts add chopped carrot, onion, potato, celery and 1 teaspoon salt. Sweat vegetables until softened, but still have a bite to them, about 8 minutes. If they start to brown, reduce the heat a bit.
Add flour to vegetables, stir to combine and cook for 1 minute. Slowly whisk in dry sherry, chicken stock and milk. Quickly stir the liquid with the vegetables to make sure no lumps form. Once the flour, veggies and liquid are incorporated, bring the mixture up to a boil and then reduce to a simmer. Simmer until the mixture has thickened significantly.
Turn the heat off and add in the cheese, while stirring. a little bit at a time until it's all incorporated. Add diced chicken. Stir to combine. Season to taste with salt and pepper. For my taste, I added about another 1/4-1/2 tsp salt.
Add frozen peas to the filling and toss to incorporate.
Transfer to prepared baking dish and top with biscuits. Brush with egg wash mixture. Bake in the oven for 20-25 minutes or until biscuits are golden brown and cooked through and the filling is hot and bubbly. Garnish with chopped chive.
While the veggies cook and the filling simmers, make the biscuits. Whisk flour, baking powder, baking soda and salt together in a large bowl. Add in chives and cheese. Stir with a fork to combine. Add in COLD cubed butter and shortening. Use a pastry cutter or fork to cut the fat in to the flour mixture. It shouldn't take long, you want there to be small, pea-sized chunks of fat distributed throughout the flour. Some larger pieces of butter are okay.
Once the butter and shortening are cut in, drizzle half of the buttermilk over the flour, chive and cheese mixture. Use your fork to gently toss until evenly coated. Add the other half of the buttermilk and continue to stir until incorporated. DO NOT OVER MIX! There will be a few shaggy pieces, that's okay.
Dump dough out on to a lightly floured surface. Gently knead the dough to bring it together completely. Lightly dust a rolling pin and roll the dough into a 6x8 rectangle. Fold the dough in half over itself and roll into another rectangle the same size. Repeat two more times. The last time you roll the dough into a rectangle, continue to roll the dough out so it's about 1/4 inch inch tall.
Use a 2 1/2-inch biscuit cutter to cut 12 biscuits. If needed, reform scraps of dough to make all 12. Make sure you don't twist the biscuit cutter into the dough, just go straight down forcefully and straight back up. To prevent the cutter from stick, dip in flour before you make each biscuit. Place the biscuits on a small baking sheet and place in the freezer until the filling is complete.