In a zip-top bag, combine bread flour, whole wheat flour, salt and butter. Place in freezer for 1 hour.
Place a 10-12" cast iron skillet over high heat. Add 1 tbsp vegetable oil. When it shimmers, cook chicken pieces (in batches, if necessary to avoid crowding) along with 1 tsp kosher salt. When cooked through but not browned, transfer to a large bowl and set aside.
Return pan to heat, but reduce to medium. Add another 1 tbsp oil, then add onions, mushrooms, carrots, celery and garlic, as well as 1 1/2 tsp kosher salt, black pepper, dried thyme and dried tarragon. Cook, stirring often, until onions are wilted, about 4 minutes. Add butter. When melted, sprinkle over flour and stir well for 1-2 minutes to cook out the raw-flour taste. Slowly stir in chicken broth and milk. Bring to a simmer and cook 3 minutes until thickened.
Off heat, return chicken to sauce, along with peas, fresh thyme and fresh tarragon. Place a large flat lid on top of the skillet and set aside.
Dump the flour-butter mixture onto a clean counter. Using a bench scraper, chop into the mixture until the butter is reduced to the size of peas. Sprinkle ice water over the mixture 1-2 tbsp at a time, then fold in with bench scraper. Once 8-10 tbsp of water is incorporated, and the flour is starting to form clumps, scrape the mixture together into a 4" square pile, and start whacking it with a rolling pin, until it forms a 12x4" rectangle. Fold in thirds and repeat. If the butter starts to feel soft, place a baking sheet (which you previously stored in the freezer) on top, then place ice packs or zip-top bags of ice on the sheet pan, for 5-6 minutes, until it firms up again.
After 4 repeats of the letter fold above, the last will be a book fold - fold both ends into the center, then fold in half at the resultant seam. Wrap tightly in parchment paper and refrigerate 1 hour.
Preheat oven to 425°F.
Set 4 16-ounce souffle dishes or onion soup crocks on a baking sheet. Beat together egg wash and have standing by. Cut dough in half; roll one piece out to about 1/4" thick, using bench flour as necessary to avoid sticking. Find a pot lid a little wider than your dishes and use it to trace out two circles from the dough; repeat with the other piece. Brush each disk with egg wash.
Fill dishes almost to the top with chicken mixture. Top each with a disk of dough, egg wash side down. Brush the exposed tops with more egg wash. Bake in preheated oven 15 minutes, then rotate pan 180°, lower heat to 400F and cook another 15-20 minutes until tops are deep brown and crispy. Cool on a rack 10-15 minutes before serving.