Cook the Pasta: In a large pot of salted boiling water, cook the pasta according to package instructions until al dente. Drain and set aside.
Sauté the Vegetables: In a large skillet, heat the olive oil over medium heat. Add the diced onion and sauté for about 5 minutes until translucent. Stir in the minced garlic and cook for another minute until fragrant.
Add Chicken and Vegetables: Add the cooked chicken, diced carrots, and celery to the skillet. Cook for about 5-7 minutes, stirring occasionally, until the vegetables start to soften.
Make the Sauce: Sprinkle the flour over the chicken and vegetable mixture, stirring to coat. Gradually pour in the chicken broth while stirring continuously to avoid lumps. Bring the mixture to a simmer, then add the heavy cream, thyme, rosemary, salt, and pepper. Cook for an additional 5 minutes until the sauce thickens.
Combine with Pasta: Add the cooked pasta and frozen peas to the skillet, stirring to combine everything evenly. Cook for another 2-3 minutes until heated through.
Prepare the Puff Pastry (Optional): If using puff pastry, preheat the oven to 400°F (200°C). Roll out the puff pastry on a lightly floured surface and cut it into small squares or shapes. Place them on a baking sheet lined with parchment paper and bake for 15-20 minutes, or until golden and puffed.
Serve: Spoon the Chicken Pot Pie Pasta into bowls and top with the crispy puff pastry pieces if desired. Garnish with fresh parsley before serving.