Position a rack in the middle of the oven. Heat oven to 425ºF.
Make the filling: Place 1 tablespoon vegetable oil into a 10-inch cast-iron skillet and set over high heat until it shimmers. Add the chicken, season with 1 teaspoon of salt, and cook, stirring occasionally, until the chicken is no longer pink on the outside but not dry, 4 to 6 minutes. Evacuate the chicken to a medium bowl and set aside.
Decrease the heat to medium and heat the remaining tablespoon vegetable oil in the skillet until it shimmers. Add the onions, mushrooms, carrots, celery, garlic, remaining 1 1/2 teaspoons of salt, black pepper, dried thyme, and dried tarragon, and stir to combine. Cook, stirring occasionally, until the onions have softened, about 5 minutes. Add the butter and melt. Stir in the flour and cook for 1 to 2 minutes. Gradually stir in the chicken broth and milk. Bring to a simmer, stirring continually. Continue stirring until the sauce thickens, about 3 minutes. Remove the skillet from the heat and stir in the peas, thyme, tarragon, and reserved chicken. Cover and set aside.
Make the crust: Combine the bread flour, whole wheat flour, and salt in a medium mixing bowl. Place in the freezer for 1 hour.
Pour the flour mixture and butter onto a clean work surface and form a mound. Use a bench scraper to cut the butter into the flour until the mixture is crumbly and the butter pieces are about the size of almonds. Add the ice water, 2 tablespoons at a time, and mix into the dough using the bench scraper until it just barely comes together. Do not add too much water, it will come together.
Shape the dough into a rectangle and pound with a rolling pin until it is about the size of a sheet of notebook paper, approximately 8 1/2 by 11 inches.
Use the bench scraper to fold the dough into thirds, like a letter. Pound with the rolling pin until the piece of dough is again the size of a sheet of notebook paper and rotate 90 degrees. Repeat the pounding, folding, and rotating 2 more times for a total of 4 turns of the dough.
Pound with rolling pin and fold the shorter sides of the rectangle in towards the center, from top to bottom, and then fold in half, like a book. Wrap in parchment paper and refrigerate for 1 hour.
Combine the beaten egg and water in a small dish and set aside.
Build the pie: Divide the warm filling into 4 (16-ounce) ramekins and place on a half-sheet pan lined with parchment paper.
Remove the puff pastry from the refrigerator. Divide in half and return one half to the refrigerator. Sprinkle the remaining half lightly with flour and roll out 1/4-inch thick. Using your ramekins as a guide, cut 2 circles that are 1/2 inch wider than the rim using a sharp knife or pizza cutter. Repeat with the other half of the dough. Refrigerate excess dough for another use.
Brush each dough round with egg wash and place egg washed-side down onto the top of each filled ramekin. Brush the top of the pastry with egg wash. Bake for 15 minutes. Decrease the heat to 400ºF and bake until the crust is puffed and golden brown and the filling is bubbly, another 15 to 20 minutes. Cool on a rack for 10 minutes before serving.