Preheat the oven to 200°C (390°F) and prepare your pie dish by setting it aside.
Melt a bit of butter in a large skillet over medium heat until it starts to smell nutty and turns slightly golden.
Add chopped onions to the skillet and cook, stirring often, until translucent and fragrant, about 5 minutes.
Stir in the minced garlic and cook for another minute until aromatic.
Add the cooked chicken, peas, and diced carrots to the skillet, stirring to combine and warm through for about 5 minutes.
Pour in the chicken broth and add a splash of cream, then bring the mixture to a gentle simmer, stirring occasionally, until slightly thickened, about 3 minutes.
Season the filling with salt, pepper, thyme, and paprika, then remove from heat and let it cool slightly.
Roll out your pie dough on a floured surface to fit your pie dish, then carefully transfer it into the dish, pressing gently to fit the bottom and sides.
Fill the pie crust with the warm chicken mixture, spreading it evenly.
Roll out the remaining dough to cover the pie, then seal the edges by pinching or crimping. Cut a few small vents on top to allow steam to escape.
Brush the top crust with beaten egg to give it a shiny, golden finish, then place the pie in the oven.
Bake for 30–35 minutes until the crust is beautifully golden and crisp, bubbling at the edges.
Let the pie rest for about 10 minutes before slicing to allow the filling to set. Serve warm and enjoy the comforting, mysterious aroma.