Preheat your oven to 200°C (390°F). Gather your pie dish and have your ingredients ready for assembly.
Melt the butter in a large skillet over medium heat until it begins to smell nutty and transparent, about 2-3 minutes.
Add chopped onions and sauté until they turn translucent and fragrant, about 5 minutes, stirring occasionally.
Stir in the minced garlic and cook for another minute until aromatic.
Add shredded cooked chicken, peas, and diced carrots to the skillet, stirring well to combine. Cook for 5 minutes until warmed through and vegetables start to soften.
Pour in the chicken broth and heavy cream, then sprinkle in smoked paprika, thyme, salt, and pepper. Bring the mixture to a gentle simmer and cook until it thickens slightly, about 3 minutes, stirring often.
Remove the skillet from heat and set aside. Roll out your pie dough on a floured surface to fit your pie dish, then gently press it into the dish, trimming excess edges.
Pour the warm filling into the prepared pie crust, spreading evenly to fill the dish.
Cover the filling with the second sheet of dough, sealing the edges by pinching or crimping. Cut a few small vents on top to allow steam to escape.
Brush the top crust with the beaten egg for a glossy, golden finish.
Place the pie in the oven and bake for 30-35 minutes until the crust is beautifully golden and crisp, and the filling is bubbling around the edges.
Remove the pie from the oven and let it rest for 10 minutes before slicing. Serve hot, enjoying the crispy crust and savory filling.