Preheat your oven to 400°F (200°C).
In a large skillet, heat the olive oil over medium-high heat. Add the chicken pieces, season with salt and pepper, and cook until browned on all sides. Remove the chicken and set aside.
In the same skillet, add the onion and garlic, sautéing until fragrant and translucent, about 3 minutes.
Stir in the carrots and celery, cooking for another 5 minutes until they start to soften.
Return the chicken pieces to the skillet and add the frozen peas. Sprinkle the flour over the chicken and vegetables, stirring to coat evenly.
Gradually pour in the chicken broth and milk, stirring continuously to avoid lumps. Bring the mixture to a simmer.
Add the orzo pasta, thyme, and rosemary. Cook for 8-10 minutes, stirring occasionally, until the orzo is tender and the mixture has thickened.
Pour the creamy chicken mixture into a large oven-proof dish.
Roll out the puff pastry and place it over the dish, trimming any excess. Press the edges to seal and create a decorative pattern if desired.
Brush the top of the puff pastry with the beaten egg.
Bake in the preheated oven for 15-20 minutes, or until the pastry is golden brown and puffed.
Remove from the oven and let it cool slightly before serving.
Garnish with freshly chopped parsley before serving.