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To make the biscuits: In a large bowl, stir the flours, baking powder, baking soda, salt, and sugar together. With a pastry cutter or your fingers, cut the butter into the dry ingredients until the butter resembles course meal with some larger butter lumps remaining. Stir in ¾ cup of the buttermilk with a fork until the mixture forms a soft, slightly sticky ball. If the dough is too dry, add a little extra buttermilk - 1 tablespoon at a time up to 2 tablespoons.
Transfer the dough to a well-floured work surface and pull it together into a ball with your floured hands. Pat the dough out to ¾-inch thickness and cut out 12 rounds with a 3-inch biscuit or cookie cutter. Transfer the biscuits to a parchment-lined baking sheet and chill until you're ready to use them. The biscuits can be made and kept covered in the refrigerator for up to 2 days in advance.
To make the filling: Preheat the oven to 400° F. Heat a large heavy-bottomed pot or Dutch oven over medium-high heat and add the oil. Once the oil is hot, sauté the onions, carrots, and celery until tender, about 5-7 minutes. Season with salt and pepper to taste. Transfer cooked vegetables to a bowl with the chicken; set aside. Reduce the heat to medium and add the butter to the same pot. When melted, stir in the flour and cook, stirring constantly for 1 minute. Whisk in the milk, chicken broth, and thyme. Allow the sauce to come to a simmer and let it simmer for 1 minute to thicken. Season with salt and pepper and stir in the sherry (if using).
Turn off the heat and transfer the vegetables and chicken to the pot, stirring until the filling is well-combined. Mix in the peas and parsley. If the filling is too thick (you'll know), stir in the extra chicken broth, about ¼ cup at a time up to ¾ cup until the filling loosens up a little.
Pour the filling into a 13×9-inch baking dish and bake for 18 minutes. Remove the dish from the oven (keeping the oven on and door closed) and top the filling with the biscuits. Return the baking dish to the oven and bake for an additional 10-12 minutes, until the biscuits are golden brown and the filling is bubbly. Cool for 5 minutes before serving.