In a large bowl, mix the flour, salt, and sugar. Add the cold butter and use a pastry cutter or your fingers to work the butter into the flour until the mixture resembles coarse crumbs.
Gradually add the ice water, one tablespoon at a time, and mix until the dough comes together. Divide the dough into two disks, wrap in plastic wrap, and refrigerate for at least 1 hour.
Prepare the Filling:
In a large skillet, melt the butter over medium heat. Add the onions and garlic and cook until softened, about 3-4 minutes.
Sprinkle the flour over the onion mixture and stir well to combine. Cook for 1-2 minutes to remove the raw flour taste.
Gradually add the chicken broth and milk, stirring constantly, until the mixture thickens and becomes smooth.
Add the cooked chicken, peas and carrots, corn, thyme, and rosemary. Season with salt and pepper to taste. Stir well and cook for an additional 2-3 minutes. Remove from heat and let cool slightly.
Assemble the Pie:
Preheat your oven to 375°F (190°C). Roll out one disk of dough on a lightly floured surface to fit your pie dish. Carefully transfer the dough to the pie dish and trim any excess around the edges.
Pour the chicken and vegetable filling into the prepared pie crust.
Roll out the second disk of dough and place it over the filling. Trim any excess dough and crimp the edges to seal. Cut a few small slits in the top crust to allow steam to escape.
Brush the top of the pie with the beaten egg.
Bake the Pie:
Bake the pie in the preheated oven for 40-45 minutes, or until the crust is golden brown and the filling is bubbling. If the edges of the crust begin to brown too quickly, cover them with aluminum foil.
Remove the pie from the oven and let it cool for about 10 minutes before serving.
Enjoy your delicious homemade savory chicken and vegetable pie!
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