Fill a 2-cup measuring cup with ice. Top measuring cup off with water (at least ½ cup). Set aside.
Cube butter into ½-inch to 1-inch pieces with varying rough chops. Toss together flour and salt in a large bowl.
Toss butter cubes into flour mixture, and separate cubes with your fingers before beginning to work together. Work in butter using your hands, making smears and buttery flakes. Do not use a machine or a pastry cutter; using your hands will help you discern the temperature and feel of the flour. You are looking to incorporate all the butter into the flour without overworking. Every piece of butter should be smeared with some flour before you begin to add the ice water; there should be no untouched pieces of butter. Doing this deftly is key; if the butter starts to feel too warm, use your refrigerator as a tool—place bowl in freezer or refrigerator at any time to cool off. Preferably, though, you will be able to work the butter in quickly without too much fuss.
To incorporate the ice water, start by drizzling about 3 tablespoons water (do not add ice) into flour mixture. Using your hands slightly cupped as paddles (do not use your fingers or do any kneading), toss the water into the flour until it is fully absorbed. Add additional water, 1 tablespoon at a time, paddling with hands between each addition, until mixture begins to form a shaggy dough. You will likely use about ¼ cup to ⅓ cup water total, depending on the humidity. The dough should feel moist but not wet or sticky. Turn dough out onto a clean work surface. Give dough a few strong kneads to work it all together (about 4 turns). Your dough should feel tacky and supple but not sticky.
Form each into a flat disk and wrap in plastic wrap. Chill in refrigerator at least 30-minutes, or up to 3-days.
Pie Filling
Preheat oven to 400°F
While the dough is resting, heat a 12-inch cast iron skillet over medium high heat. Add 2 to 3-tablespoons of neutral cooking oil.
Generously salt your chicken breasts and add to the pan. Cook for approximately 8 to 10-minutes per side until the outside is golden brown and the internal temperature reaches 165°F. Remove the chicken from the pan and allow to cool.
Add your onion with a pinch of kosher salt and cook for 5 to 7-minutes, stirring occasionally, until softened. Add more oil if needed.
Add the chicken stock to deglaze, scraping up any browned bits from the bottom of the pan.
While stirring constantly, sprinkle in flour to evenly coats the onions. Cook for 2 to 3-minutes.
While whisking/stirring constantly, slowly add in your measured kosher salt, milk and heavy cream. Bring to a boil, and reduce heat to medium-low. Simmer for 8 to 10-minutes, stirring occasionally until sauce is thickened, and begins to leave a bare spot in the pan when a spoon is dragged through it.
While sauce is simmering, use two forks to shred (or your hands to pull) cooked chicken into bite-sized pieces. Rough chop the chicken if the pieces are too big after pulling/shredding, Do not drain any of the juices that have collected.
Remove the sauce from heat and add your frozen peas and carrots, shredded chicken (and juices), and fresh thyme.
Season to taste with kosher salt and pepper. Allow to cool for several minutes.
Assembly
While filling and pan is cooling, remove pie crust from the fridge. Unwrap and roll out on a floured work surface until ¼-inch thick with a floured rolling pin.
Pick up the pie crust and drape the it over the pan, allowing the dough to extend past the edges. Gently press the dough into the corners of the pan to completely cover the filling.
Starting in the middle of the pan, press down with your rolling pin and roll it over the top of the skillet to "cut" the excess pie dough off. Discard or freeze your excess dough (there will be extra).
Brush your egg wash mixture evenly over your pie dough (there will be excess egg wash - just discard it when done). Cut two to three slits in the center of the dough to allow steam to vent.
Bake for 40 to 45-minutes until the filling is bubbling around the sides and the crust is golden brown.
Remove and allow to cool for 10 to 15-mintues before serving. Garnish with flaky salt and fresh thyme.
Notes
The filling can be made on the stove top and then transferred to a 9x13-inch baking dish. Just roll the dough into a rectangle shape and follow the covering/baking directions as written.