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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.
Add chilled diced butter to the bowl. On the lowest speed, turn the mixer on and off quickly for a few seconds to coat the butter with the flour. This helps to prevent the flour from spilling over. Continue to mix on low speed until the flour and butter resemble wet sand with coarse crumbles and some pea-sized pieces remaining, about 60 to 75 seconds. Use 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.
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.
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.
Remove the crusts from the refrigerator, 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.
Cut each chicken breast in half through the center to create two cutlets. Lightly season both sides with salt and black pepper.
Set the oven rack to the middle position. Preheat to 375ºF (190ºC). Heat a large skillet with high sides or dutch oven over medium heat. Once hot, add the olive oil. Add the chicken and cook for 6 minutes. Flip over 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.
Rest the chicken until it’s cool enough to handle. Shred into smaller pieces and set aside.
Heat the same pan used to cook the chicken over medium heat. Melt the butter in the pan. 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.
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.
Transfer the filling to a sheet pan, spread into a thin layer, and allow 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.
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.
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.
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.