Start by preheating your oven to 200°C/400°F/Gas Mark 6.
Using a large frying pan, heat up the olive oil over medium heat. Once hot, add the chopped onion and minced garlic to the pan and cook for 2-3 minutes until softened.
Next, add the chicken to the frying pan, and cook for 5-7 minutes, stirring occasionally, until browned all over.
Add the diced carrots and celery to the same pan as the chicken and cook for 5 minutes until slightly softened.
In a separate saucepan, melt the butter over medium heat. Once melted, add the plain flour, and stir constantly for 1-2 minutes to create a roux.
Gradually whisk in the chicken broth and whole milk to the roux, stirring continuously until the sauce is smooth. Add the dried thyme, salt, and black pepper, and continue to whisk until the sauce has thickened.
Add the sauce to the chicken and vegetables in the frying pan and stir to combine.
Transfer the chicken mixture to a large ovenproof dish.
Roll out the puff pastry sheet to fit the size of the dish and place it on top of the chicken mixture. Cut a few slits in the pastry to allow steam to escape.
Brush the beaten egg over the pastry to glaze.
Bake the chicken pot pie in the preheated oven for 25-30 minutes, or until the pastry is golden brown and puffed up.
Allow the pie to cool for a few minutes before serving. Enjoy your delicious chicken pot pie!