Make the Dough: In a food processor, pulse the flour, sugar, and salt. Add the cold, cubed butter and pulse briefly until the mixture looks like coarse crumbs (pea-sized chunks of butter should remain). Drizzle in the ice water while pulsing just until the dough begins to clump together.
Chill: Divide the dough into two disks, wrap tightly in plastic wrap, and refrigerate for at least 1 hour. Tip: Cold dough is the secret to flaky layers.
Cook the Vegetables: While the dough chills, melt 6 tablespoons of butter in a Dutch oven over medium heat. Add onions, carrots, and mushrooms. Sauté for 8-10 minutes until softened. Add the garlic and thyme, cooking for 1 minute until fragrant.
Make the Roux: Sprinkle the flour over the vegetables and stir constantly for 2 minutes to cook out the raw flour taste.
Create the Gravy: Slowly whisk in the chicken broth and heavy cream. Bring to a simmer and cook for 2-3 minutes until thickened to a gravy consistency.
Season and Cool: Stir in the salt and pepper. Taste test now. If your broth was salty, you may need less salt. Stir in the cooked chicken, peas, and parsley. Remove from heat and let the filling cool completely to room temperature. Note: putting hot filling in a raw crust will cause a soggy bottom.
Preheat: Preheat oven to 425°F (220°C). Place a heavy baking sheet on the middle rack to heat up (this helps crisp the bottom crust).
Roll and Fill: On a floured surface, roll out the first dough disk and line a 9-inch pie plate. Pour the cooled filling into the crust. Roll out the second disk and place it over the top.
Seal and Wash: Trim excess dough and crimp the edges to seal. Cut 4-5 slits in the top for steam to escape. Whisk the egg with 1 tablespoon of water and brush over the top crust. Sprinkle with flaky sea salt.
Bake: Carefully place the pie dish onto the hot baking sheet in the oven. Bake for 35-40 minutes until the crust is deep golden brown and the filling is bubbling through the vents. If the edges brown too fast, cover them with a foil shield.
Rest: Let the pie rest for at least 15-20 minutes before slicing. This allows the filling to set so it doesn't run when you cut it.