Combine flour and salt in a large bowl. Cut in the cold butter using a pastry cutter or food processor until the mixture resembles coarse crumbs with pea-sized butter pieces visible.
Gradually add ice water mixed with apple cider vinegar, stirring just until the dough comes together. Divide into two portions (one slightly larger for the bottom crust), shape into discs, wrap in plastic, and refrigerate for at least 30 minutes.
Heat olive oil in a large skillet over medium heat. Add onions, carrots, and celery, sautéing until the onions become translucent (about 5-7 minutes).
Add minced garlic and cook for another minute until fragrant.
Sprinkle flour over the vegetables and stir constantly for 2 minutes to cook out the raw flour taste.
Gradually add chicken broth while stirring continuously, followed by milk. Bring to a gentle simmer and cook until the sauce thickens enough to coat the back of a spoon, about 5-7 minutes.
Fold in the shredded chicken, peas, and herbs. Season with salt, pepper, and nutmeg. Allow the mixture to cool slightly before filling the pie.
Preheat your oven to 375°F (190°C). Roll out the larger disc of dough on a floured surface to about 12 inches in diameter for a 9-inch pie dish. Transfer carefully to your pie dish, allowing the excess to hang over the edges.
Pour in the filling and roll out the second disc for the top crust. Place over the filling, trim the excess, and crimp the edges to seal.
Cut 4-5 slits in the center to allow steam to escape, and brush with beaten egg for a golden, glossy finish.
Bake for 40-45 minutes until the crust is deeply golden and the filling is bubbling through the vents. Place a baking sheet beneath the pie to catch any potential overflow.