In a large bowl, whisk together the flour, salt, and sugar. Add the cold cubed butter and use a pastry cutter or your fingertips to work it into the flour until the mixture resembles coarse crumbs with visible pea-sized butter chunks remaining.

Drizzle in ice water one tablespoon at a time, stirring with a fork after each addition, until the dough just comes together when pressed. It should not feel wet or sticky.

Turn the dough out onto a lightly floured surface, divide in half, and form each half into a disk. Wrap in plastic wrap and refrigerate for at least 1 hour, or up to 2 days.

Place the chicken breasts in a medium saucepan and pour over 3 cups of the chicken broth. Bring to a gentle simmer over medium heat, then reduce to low and cook for 15-18 minutes until the chicken is cooked through and no longer pink in the center.

Transfer the chicken to a cutting board and let it rest for 5 minutes. Shred or chop into bite-sized pieces. Reserve the poaching broth in the saucepan.

Preheat your oven to 400°F (200°C). Lightly grease a 9-inch deep-dish pie pan and set aside.

In a large skillet or Dutch oven, melt the butter over medium heat. Add the onion, carrots, and celery and cook, stirring occasionally, for 6-8 minutes until softened. Add the garlic and thyme and cook for 1 minute more until fragrant.

Add the diced potatoes to the skillet and stir to combine. Sprinkle the flour evenly over the vegetables and stir constantly for 1-2 minutes to cook out the raw flour taste.

Slowly pour in the reserved poaching broth plus the remaining 1 cup of broth, whisking or stirring constantly to prevent lumps. Add the milk or half-and-half and stir to combine.

Bring the mixture to a simmer and cook for 8-10 minutes, stirring frequently, until the potatoes are just tender and the sauce has thickened enough to coat the back of a spoon.

Remove from heat and stir in the shredded chicken, frozen peas, parsley, apple cider vinegar, salt, and pepper. Taste and adjust seasoning generously — the filling should taste boldly seasoned. Let it cool for 10 minutes.

On a lightly floured surface, roll out one disk of pie dough into a 12-inch circle. Carefully drape it into the prepared 9-inch deep-dish pie pan, pressing it gently into the bottom and sides and letting it hang over the edges by about 1 inch.

Pour the cooled filling into the bottom crust, spreading it into an even layer.

Roll out the second disk of dough into a 12-inch circle and lay it over the filling. Trim any excess dough, then crimp the edges together firmly to seal. Cut 4-5 slits in the top crust to allow steam to escape.

Brush the top crust evenly with the egg wash for a golden, glossy finish.

Place the pie on a rimmed baking sheet to catch any drips and bake in the preheated 400°F oven for 40-45 minutes, until the crust is deep golden brown and the filling is bubbling through the slits.

Remove from the oven and let the pot pie rest for at least 15 minutes before slicing and serving. This resting time is essential — it allows the filling to set up so it holds its shape when sliced.