Position an oven rack in the middle of the oven and preheat to 425 degrees F.
For 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. Remove the chicken from the skillet and set aside in a medium bowl.
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.
For the crust: Combine the bread flour, whole wheat flour, and salt in a medium mixing bowl. Place in the freezer for 1 hour.
Place the flour mixture and butter in a mound on a clean work surface. 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 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.
For the egg wash: Combine the beaten egg and water in a small dish and set aside.
To 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. Return one half to the refrigerator. Sprinkle lightly with flour and roll to 1/4-inch thick and, 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 400 degrees F 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.