Heat olive oil in a large skillet over medium heat.
Add diced chicken and cook until browned and cooked through. Remove from the skillet and set aside.
In the same skillet, add onion, garlic, carrot, celery, and mushrooms. Sauté until vegetables are tender.
Return the chicken to the skillet. Add chicken broth, heavy cream, and thyme. Season with salt and pepper.
Simmer for 10-15 minutes until the mixture is well combined. If the sauce is too thin, sprinkle in flour and stir until thickened. Let the filling cool slightly.
Assemble the Cake:
Preheat your oven to 375°F (190°C).
Roll out one sheet of puff pastry and place it in a pie dish or baking dish, leaving some overhang.
Pour the cooled chicken fricassee filling into the pastry-lined dish.
Roll out the second sheet of puff pastry and place it over the filling. Seal the edges by crimping with a fork or your fingers.
Cut a few slits on top to allow steam to escape.
Brush the top with beaten egg for a golden finish.
Bake:
Bake in the preheated oven for 25-30 minutes or until the pastry is golden brown and puffed.