Step 1: To make the pie crust, whisk the flour, sugar, and salt in a large bowl. Then, add the cubed butter and toss to coat. Next, onto a clean surface, dump the mixture. Roll the butter into thin sheets using a rolling pin, mixing it with the flour. Scrape the rolling pin using a bench scraper, bringing the mixture back into a pile as needed. Do this until the butter is combined into the flour. Place the mixture back into the bowl and chill in the fridge for about 15 minutes.
Step 2: Add the buttermilk to the mixture in the bowl and mix using your hands until it forms a ball. Add a tbsp of water at a time if the mixture is too dry. Split the dough into two, then flatten each into a disk. Using plastic wrap, wrap each disk and place it in the fridge to chill.
Step 3: For the filling: In a large skillet, heat the butter over medium-high heat. Once the butter has melted, add the onions, carrots, celery, and garlic to the skillet and saute until tender. Add in the flour, salt, black pepper, thyme, parsley, chicken broth, and heavy cream. Stir until the lumps are gone, then simmer for about 10 minutes over medium-low heat or until the sauce has thickened. Then, add the shredded chicken or turkey and frozen peas. Give it a good stir, then take off the heat and set aside.
Step 4: Prepare the oven. Preheat it to 400 degrees F.
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