Preheat your oven to 200°C (390°F). This ensures it's ready for baking once the filling is prepared.
In a large skillet, melt a bit of butter over medium heat until it starts to smell nutty and becomes fragrant.
Add chopped onions to the skillet and cook, stirring often, until they become translucent and soft, about 5 minutes, filling your kitchen with a sweet aroma.
Stir in the minced garlic and cook for another minute until fragrant, avoiding browning.
Gently fold in the chopped cooked chicken, peas, and carrots, cooking for about 5 minutes until everything is warmed through and the mixture begins to bubble.
Pour in the chicken broth and a splash of cream, then bring the mixture to a simmer. Cook for about 3 minutes until the filling thickens slightly and turns velvety.
Season with salt, pepper, thyme, a dash of smoked paprika, and stir in freshly chopped parsley for a bright finish. Remove from heat and set aside.
On a floured surface, roll out the pie dough with a rolling pin until it’s large enough to fit your pie dish, about 1/8 inch thick.
Transfer the rolled dough into the pie dish, gently pressing it into place to avoid air pockets.
Fill the crust with the warm chicken mixture, spreading it evenly with a spoon or spatula.
Roll out the remaining dough to create the top crust, then cover the filling, sealing the edges by pinching or crimping.
Cut a few small vents on the top crust to release steam, then brush the entire surface with the beaten egg for a gorgeous, golden finish.
Bake the pie in the preheated oven for 30–35 minutes, until the crust is deep golden and crispy, with bubbling filling visible through the vents.
Remove the pie from the oven and let it rest for 10 minutes to set the filling before slicing and serving hot, enjoying its comforting, haunting aroma.