Prepare the Crust - Keep the measured ice water and diced butter in the refrigerator until ready to use. In a stand mixer bowl, add the flour and salt. Combine using the paddle attachment on the lowest speed (Stir) for about 10 seconds.
Cut the Butter into the Flour - Add chilled diced butter to the bowl. On the lowest speed, turn the mixer on and off quickly for a few seconds to coat. This helps to prevent the flour from spilling over. Continue to mix on low speed until the mixture resembles wet sand with coarse crumbles and some pea-sized pieces remaining, about 60 to 75 seconds. Use your fingers to break up any large pieces. Do not overmix. The dough should not bind together before the water is added. Alternatively, use a dough/pastry blender or your fingers to break the butter into the dough.
Hydrate the Dough - Gradually add 1 tablespoon of ice-cold water to the bowl. After each addition, turn the mixer on for 1 to 2 seconds. Only add enough water until the dough looks lumpy and hydrated, but not wet or sticky. Where it just begins to clump together with small crumbles on the bottom of the bowl. All of the water may not be needed, about 8 to 10 tablespoons is typical. When the dough is pinched together, it should compress and hold, not be dry or crumbly. Do not over mix. The dough will be pressed together before resting.
Chill the Dough - Separate the dough into 2 even-sized portions, about 1 pound (454 grams) each. Press them into a 1-inch thick round disc and cover separately in plastic wrap. Place both in a resealable plastic bag. Store in the refrigerator to rest for at least 4-hours, overnight, or up to 2 days.
Roll Out the Dough - Remove the crusts from the refrigerator and allow them to sit at room temperature for about 5 to 10 minutes or until pliable. Dust your working surface dough with flour. When rolling out, dust again if needed to prevent sticking. Roll the top crust into a 13-inch circle, about ¼-inch thick. Lay it flat on a sheet pan lined with parchment paper. Roll out the bottom crust using the same steps above. Place into a 9-inch pie dish and gently press against the sides and bottom. With a paring knife, trim the excess that hangs over the edges. Transfer both crusts to the refrigerator to chill while making the filling.
Chicken and Vegetable Filling
Prepare the Chicken - Cut each chicken breast in half through the center to create two cutlets. Lightly season both sides with salt and black pepper.
Saute the Chicken - Set the oven rack to the middle position—preheat to 375ºF (190ºC). Heat a large skillet with high sides or a dutch oven over medium heat. Once hot, add the olive oil. Add the chicken and cook for 6 minutes. Flip and cook until the internal temperature reaches 160 to 165ºF (71 to 74ºC), about 5 to 7 minutes. Turn off the heat and transfer to a clean plate—reserve the pan to cook the rest of the filling.
Shred the Chicken - Rest the chicken until it’s cool enough to handle. Shred into smaller pieces and set aside.
Cook the Vegetables - Heat the same pan used to cook the chicken over medium heat. Melt the butter. Add the onions and garlic, saute until translucent, 2 minutes. Add the celery, carrots, rosemary, and thyme, saute for 3 minutes. Add the mushrooms, saute for 1 minute. Add the potatoes, 1 ½ teaspoons salt, and ½ teaspoon black pepper. Stir, then cover and cook until the potatoes are tender, about 9 to 10 minutes. Make sure to stir the vegetables every 2 minutes to prevent burning.
Thicken the Mixture - Sprinkle in the flour, stir and cook for 2 minutes. Slowly pour in the chicken stock and heavy cream, stir constantly until a thick and smooth sauce forms, about 2 to 3 minutes. It should coat the back of a spoon. Turn off the heat. Add the shredded chicken, peas, and chopped parsley, stir to combine.
Cool the Filling - Transfer the filling to a sheet pan, spread it into a thin layer, and allow it to cool until room temperature, about 20 minutes. Stir occasionally to help it cool faster. The mixture should not be hot when added to the pie crust. The filling can also be transferred to an airtight container once cooled and then refrigerated for up to 2 days before using.
Assemble the Pie - Remove the top and bottom crusts from the refrigerator. Evenly spread the chicken filling into the pie dish with the bottom crust. Lay the top crust over the chicken filling. Trim the excess dough, to about ½-inch hanging over the pie plate. Fold the excess underneath the bottom crust edges. Crimp by pinching the dough using the pointer and thumb fingers.
Bake the Pie - Lightly brush the whisked egg on the top of the crust and edges. Cut six, ¾-inch vent slits in the center for steam to escape. Sprinkle with salt and pepper. Place the pie dish on a foil-lined baking sheet. Bake until the crust is golden brown, about 60 to 70 minutes. The sides will continue to darken during cooling.
Cool and Serve - Let the pie sit on the baking sheet for 5 minutes. Cool on a wire rack until warm or room temperature, at least 45 minutes before serving. Cut and serve while still warm.
Notes
Using Salted Chicken Stock: Reduce the kosher salt input to 1 teaspoon instead of 1 ½ teaspoon. Taste and add more salt as needed.
Make-ahead: A fully assembled and unbaked pie can be made 1 day ahead. Make sure that the filling is cold before adding it to the crust. Wait to brush with egg wash until ready to bake.
Storing: The pie can be covered and refrigerated for up to 7 days.
Reheating: For a quick reheat, microwave slices on high power in 30-second increments until the filling is hot, about 2 to 3 minutes. Transfer to a foil-lined sheet pan. Broil on the lower rack of the oven until the top crust is hot and crisp. Check every minute to make sure the crust doesn't burn. Slices can also be warmed in the oven at 300ºF (149ºC) until the filling and crust are hot, about 15 to 20 minutes.
Freezing Unbaked Pie: Wrap well with plastic wrap, then foil. Freeze for up to 3 months. When ready, bake straight from the freezer—no need to defrost.
Freezing and Reheating a Baked Pie: Cool entirely, then wrap in plastic and foil for up to 3 months. Bake at 375ºF (190ºC) to reheat until warmed through. Cover the surface with some foil if it starts getting too brown.