Preheat the oven to 200°C (180°C fan).
Heat the butter and olive oil in a large frying pan and brown the chicken on all sides. Remove and set aside.
In the same pan, cook the onion until softened, then add the garlic and mushrooms. Cook until the mushrooms release their moisture and become lightly browned.
Stir in the flour and cook for 1 minute before gradually whisking in the chicken stock until smooth.
Add the double cream, Dijon mustard, thyme, parsley, nutmeg (if using), salt, and black pepper. Simmer until the sauce thickens.
Return the chicken to the pan and stir until well coated. Remove from the heat and allow the filling to cool slightly.
Spoon the filling into a pie dish and cover with the puff pastry. Trim the edges, seal, and brush with beaten egg.
Cut a few small slits in the pastry to allow steam to escape.
Bake for 25–30 minutes, or until the pastry is puffed, crisp, and golden brown.
Rest the pie for 5–10 minutes before serving.