Preheat the oven to 400°F (200°C).
Heat the olive oil in a 10-inch cast iron skillet over medium-high heat. Add the diced chicken breast, season with half of the salt and black pepper, and sear until golden brown and cooked through, about 5 to 7 minutes. Transfer the chicken to a plate and set aside.
Reduce the heat to medium. Melt the butter in the same skillet. Add the yellow onion, carrots, celery, and optional mushrooms. Sauté for 6 to 8 minutes until the vegetables are tender.
Stir in the minced garlic and cook for 1 minute until fragrant.
Sprinkle the all-purpose flour over the vegetables and stir continuously for 2 minutes to cook the raw flour taste out, forming a roux.
Gradually pour in the chicken broth while whisking constantly to prevent lumps. Stir in the heavy cream. Bring the mixture to a gentle simmer and cook for 3 to 4 minutes until the sauce has thickened.
Remove the skillet from the heat. Stir in the cooked chicken, frozen peas, fresh parsley, fresh thyme, and the remaining salt and black pepper.
Unroll the puff pastry sheet and carefully drape it over the skillet, trimming or folding any overhanging edges. Cut 3 or 4 small slits in the center of the pastry to allow steam to escape.
Brush the entire surface of the puff pastry with the beaten egg.
Bake in the preheated oven for 20 to 25 minutes, or until the puff pastry is deep golden brown and crisp, and the filling is bubbling visibly around the edges.
Remove from the oven and let the pot pie rest for 10 minutes before serving.