In a large mixing bowl, combine all-purpose flour, granulated sugar, and salt. Cut in very cold butter using a pastry cutter until the mixture resembles coarse crumbs. Gradually add ice water, mixing until the dough comes together. Divide the dough into two balls, wrap in plastic wrap, and refrigerate for 1 hour.

Preheat your oven to 425°F (220°C). Roll out one dough ball on a floured surface into a circle to fit your pie plate. Place the rolled-out dough into the pie plate, pressing gently against the sides. Return it to the refrigerator while preparing the filling.

In a medium saucepan, fill with water and bring to a boil. Add chopped carrots, frozen corn, and frozen peas, cooking until tender, about 15 minutes. Drain and set aside to cool.

In a large pot over medium heat, melt unsalted butter. Add diced yellow onion, chopped celery, and minced garlic, sautéing until the onion is translucent. Sprinkle in all-purpose flour, stirring until golden, then gradually whisk in chicken broth and whole milk until thickened, about 5 minutes. Stir in rotisserie chicken and cooked vegetables.

Pour the creamy filling into the chilled crust, spreading evenly. Roll out the second dough ball and place it over the filling, crimping the edges to seal. Cut slits in the top crust for steam to escape.

Whisk egg and water for an egg wash. Brush the mixture over the top crust. Place in preheated oven and bake for 30-35 minutes, or until golden brown and bubbly. Let cool for a few minutes before slicing.