Make the dough: Put the flour and salt into a large food processor. Pulse them together until the flour is fluffy and aerated.
Add the butter. Pulse ten times. The flour and butter should look like a sandy mixture with chunks of butter still visible. Pour in 1/2 cup of the water. Pulse five times. The dough should look moistened but not wet, and gathering but not collected into a ball. If there are any dry patches of flour remaining, add more water 1 tablespoon at a time. The final dough should look a bit like cheese curds.
Form the dough into discs: Dump the curds of dough onto a clean, cool surface. (A piece of parchment paper works well here.) Gather them loosely into a pile. Put the heel of your hand on the top portion of the pile and press down and away from you. Repeat this with the entire pile until you have a smooth pie dough.
Cut the pie dough in half. Take one half and press the dough onto itself, as though you are closing the page of a book; repeat with the other side. Form the pie dough into a smooth disc and cover with plastic wrap. Repeat with the other half of the dough. Let the discs of dough rest in the refrigerator for 30 minutes.
Roll out the dough: Take the pie dough disks out of the refrigerator and let them sit out until they have come to a chilly room temperature, about 15 minutes or so. Put one disc of dough between two lightly greased pieces of parchment paper. Pat down the disk and put the rolling pin on top of it. Now, imagine that the dough is the face of a clock. Roll out once at 12 o'clock. Then, lift the pin and roll at 12:10. Moving in "ten-minute" increments, roll out the dough to slightly larger than your pie pan.
Lift the top paper, put a 9-inch pie pan on top of the pie dough, and flip it over. Carefully, strip away the parchment paper. Go slowly. Voila! Pat the dough down into the pan. If some of the pie dough has stuck onto the parchment, no need to worry. Simply peel it off and pat into the rest of the pie dough.
Roll out the second disc of pie dough in a similar fashion. Put the pie pan and the top piece of dough in the freezer while you make the chicken filling.
Sear the chicken: Set a Dutch oven over medium-high heat. Season the chicken with salt and pepper. Pour in 2 tablespoons of the oil. Put half of the chicken into the hot oil. Let it cook for 2 to 3 minutes, then stir it up and cook until it is evenly browned. Remove the cooked chicken from the oil and repeat with the remaining chicken.
Cook the vegetables: Heat the remaining 2 tablespoons of oil in the Dutch oven. Add the mushrooms, carrot, onion and garlic and cook, stirring frequently, until the onion is soft and translucent, about 7 minutes. Add the thyme and cook until it is fragrant, about 1 minute. Add the potato.
Make the gravy: Sprinkle the flour over the vegetables and stir. Cook for 2 to 3 minutes, stirring constantly. Pour in the chicken stock, stirring it into the vegetables. Bring the stock to a boil and then turn the heat down to simmer. Simmer the stock, stirring occasionally, until it begins to thicken, about 15 minutes.
Finish the filling: Add the asparagus and edamame to the vegetable mixture. Stir until they are fully incorporated. Taste the filling and season with salt and pepper if needed.
Fill the pie: Let the filling cool to room temperature. Pour it into the prepared pie pan. Allow the remaining piece of pie dough too come to a chilly room temperature.
Top the pie: Preheat the oven to 425 degrees F. Flip the pie dough on top of the pie. If the dough sticks and breaks, just pat the pieces together. Working with slightly wet fingers, crimp the edges of the pie dough together.
Bake the pie: Put the pot pie onto a parchment-lined sheet pan to catch the drippings during baking. Bake the pot pie until the crust is golden brown and the filling is piping hot and starting to ooze out of the pie crust, about 45 minutes to 1 hour.
Cool the pie: Allow the pie to cool to room temperature. Serve!
Pour all the flours into a large container. (Restaurant supply stores sell large plastic containers that fit this purpose well. You could also use a large glass jar.) Shake and shake and shake harder until all the flours have become one color.