BUTTERMILK BISCUIT: Prepare the butter by quartering it lengthwise and then cutting crosswise into 1/2 inch pieces. Place butter in a bowl and put it in the freezer while you prepare the dry ingredients. Measure out the dry ingredients and pulse in a food processor fitted with a metal blade until well combined. Add butter and pulse just slightly until mixture resembles very coarse cornmeal with a few larger lumps.
Transfer mixture to a medium bowl and add the buttermilk. Stir until the dough gathers into moist clumps, adding up to 2 tablespoons extra buttermilk (adding 1 tablespoon at a time) if the dough is still dry. Transfer onto a floured work surface and form into a rough ball. Roll out the dough until it is 1/2 inch thick. Using a 2 1/2 inch pastry cutter, stamp out 8 rounds of dough.
At this point, refrigerate the dough rounds on a cookie sheet covered with plastic wrap while preparing the pot pie filling recipe. When the pot pie filling is ready, arrange the dough rounds over the warm filling and proceed with the recipe for the pot pie.
POT PIE: Adjust oven rack to lower-middle position and heat the oven to 400 degrees F. Bring chicken broth to a boil in a small stock pot. Add the chicken, cover and simmer for 8-10 minutes until chicken is just done. Transfer chicken to a bowl and reserve the leftover broth.
Using the same pan, heat the oil over medium-high heat. Saute the onion, celery and carrot until tender, about 4 minutes. Add garlic and saute an additional minute. Season with salt and pepper. Shred the chicken into bite size pieces and set aside in a bowl with the sauteed vegetables.
Again using the same pan, melt the butter over medium heat. Stir in the flour and cook for 1 minute. Whisk in reserved chicken broth, milk and the thyme. Bring to a simmer to thicken sauce, about 1 minute.
Stir the chicken and vegetables into the sauce. Season to taste with salt and pepper. Stir in peas and parsley and pour into a baking dish. Immediately top with biscuits. Bake at 400 degrees for 30 minutes.