To the bowl of a food processor, add the flour, chives, herbs, and salt. Pulse to combine well. Add the cold butter. Pulse just enough to cut the butter into the flour, aiming for pea-sized pieces of butter – the flour mixture should look similar to coarsely grated parmesan cheese. Pulse in the apple cider vinegar, then pulse in 1 tablespoon of ice water at a time, just until the mixture begins to pull from the side of the bowl – this usually takes 5-6 tablespoons of water.
Transfer the mixture to a lightly floured surface. Working quickly and carefully, use your hands to press the dough into a ball and flatten into a disc. Divide into 4 equal pieces, pressing each piece into a ball and flattening into a dish. Wrap in plastic and transfer to the refrigerator to chill. Dough can be prepped up to 3 days ahead of time.
Add the butter to a large skillet over medium heat. Once melted, add in the onion, carrots, celery, and potato. Season with 1 teaspoon kosher salt and ground black pepper as desired. Stir to combine. Cook, stirring occasionally, for 5-7 minutes, until the veggies are softened and slightly browned. Stir in the garlic and fresh herbs. Cook for another 1-2 minutes, until fragrant.
Sprinkle the flour over the softened veggies, stirring to combine. Cook for 1-2 minutes, until deeply browned. While scraping the bottom of the skillet with a wooden spoon, slowly pour in the white wine. It will bubble vigorously, almost cooking all the way down very quickly. Once the wine is reduced, stir in the chicken stock and heavy cream. Bring to a simmer and cook for 4-5 minutes, stirring occasionally, until thickened slightly.
Stir the shredded chicken and frozen peas into the chicken pot pie filling. Simmer 1-2 minutes longer, then remove from the heat. Taste and season with additional kosher salt and ground black pepper as desired.
Preheat the oven to 425 degrees F, ensuring a rack is positioned in the center of the oven. Divide the pot pie filling between 4 oven-safe soup bowls. Transfer to a baking sheet and set aside. Remove the chilled pot pie crust dough from the refrigerator. Working one at a time, roll out each portion on a lightly floured surface – aim for a circle about 1-inch wider than your soup bowls and about ⅛-inch thick. Brush egg was around the rim of each soup bowl, then press the pot pie crust over top. Generously brush the surface of the crust with egg wash, then slice a small vent into the center.
Transfer the individual pot pies to the oven. Bake 30-35 minutes, or until the crust is beautifully golden brown. Remove from the oven and rest 10-15 minutes before serving.