1. In a medium saucepan over high heat, bring chicken stock to a boil. Add carrots, potatoes, and celery. Lower heat to medium and cook until vegetables are tender, 5 to 7 minutes. Drain vegetables, reserving stock; set both aside separately.
2. In a large, heavy-bottomed saucepan, melt salted butter over medium heat. Add onion and cook until golden, 6 to 8 minutes. Add mushrooms and cook 5 minutes. Add fresh thyme and 5 tbsp. flour and cook 2 minutes. Slowly add milk, whisking constantly, until combined, then add stock and cook, stirring often, until mixture thickens, 8 to 10 minutes. Season with nutmeg, 1 tsp. salt, and pepper to taste. Add parsley, chicken, cooked vegetables, and peas and divide filling evenly among 6 or 7 ovenproof containers (8 to 10 oz. each), leaving the top 1/4 in. unfilled.
3. Preheat oven to 425°. To make biscuit topping, sift remaining 2 cups flour with baking powder, baking soda, and remaining 1 tsp. salt. Using your fingers or a pastry cutter, work in unsalted butter to form a coarse meal, working quickly to keep the butter from warming up and melting into the dough. Stir in cheese and sage. In a separate bowl, whisk together egg and buttermilk and add to the flour mixture, stirring gently until a shaggy dough forms.
4. Lightly flour a counter, a rolling pin, and your hands. Divide dough into 2 balls. Roll out first ball to a 1/4-in. thickness, then use a 2 1/2-in. biscuit cutter to cut into rounds, scraping and rerolling dough as needed. Repeat with second ball.
5. Place 3 rounds of dough on each potpie, overlapping as necessary (any unused rounds can be baked on their own as biscuits). Brush dough with egg wash, put potpies on a cookie sheet lined with aluminum foil, and bake until crust is golden brown and filling is bubbling, 17 to 22 minutes.
Note: Nutritional analysis is per serving.