Remove the meat from the rotisserie chicken and shred it into bite-sized pieces. Discard the bones, skin, and cartilage.
Melt the butter in a large skillet over medium heat. Add the onion, carrots, and celery. Cook until the vegetables begin to soften, about 5–7 minutes.
Add the minced garlic and cook for about 30 seconds, until fragrant.
Sprinkle the flour over the vegetables and stir continuously for about 1 minute.
Slowly pour in the chicken broth while stirring to prevent lumps. Add the heavy cream and continue cooking until the sauce becomes smooth and thickened.
Stir in the shredded rotisserie chicken, green peas, thyme, parsley, salt, and black pepper. Cook for another 2–3 minutes until everything is heated through.
Transfer the creamy chicken mixture to a 9-inch baking dish and spread it evenly.
Place the chilled puff pastry over the filling. Trim away excess pastry and press the edges gently around the dish. Cut several small vents into the top.
Brush the pastry lightly with beaten egg for a golden, glossy finish.
Bake in a preheated 400°F (200°C) oven for about 25–30 minutes, or until the pastry is puffed and deeply golden and the filling is bubbling around the edges.
Remove from the oven and let the pot pie rest for 5–10 minutes before serving. This allows the creamy filling to settle slightly.