Prepare Oven & Pan: Preheat your oven to 400ºF (200ºC). Grease a round 9-inch pie pan with butter or cooking spray and set aside.
Thaw Puff Pastry: Let puff pastry thaw until just pliable but still cold.
Dissolve Bouillon Cube: In a heat-proof measuring jug, pour in the 1 cup of boiling water and add the bouillon cube. Stir until completely dissolved. Set aside.
Cook Onion & Garlic: In a large skillet, melt the butter over medium-high heat. Add the chopped onion and sauté for 3 minutes. Add the minced garlic and sauté for 30 seconds until fragrant.
Add Carrots & Celery: Add the chopped carrots and celery. Cook on medium-high heat for about 4-5 minutes, or until slightly softened.
Stir in Flour: Sprinkle the flour over the vegetables and stir until they are well coated.
Add Liquids & Season: Pour in the prepared chicken broth and the milk. Add dried or fresh thyme, salt and pepper. Stir and allow the mixture to come to a gentle boil. Allow it to boil gently, not rapidly, for 2-3 minutes until thickened, mixing most of the time to avoid it sticking on the bottom. Taste test and season with more salt and pepper if needed.
Add Chicken & Peas: Once thickened, reduce the heat to low and stir in the shredded chicken and frozen peas. Remove from the heat.
Transfer Filling: Transfer the filling into the greased pie dish.
Prepare Egg Wash: In a small bowl, whisk together the egg and milk until fully combined.
Add Puff Pastry: Lay thawed puff pastry over the filling, trimming if needed to fit the dish. Cut a small slit in the centre to allow steam to escape.
Brush with Egg Wash: Brush the entire surface of the pastry evenly with the egg wash.
Bake the Casserole: Bake for 30-35 minutes, or until the puff pastry is golden brown and the filling is bubbly.
Rest & Serve: Let it rest for 2-3 minutes before serving. Enjoy!