Cook the Pasta:
Bring a large pot of salted water to a boil. Add penne pasta and cook according to package instructions until al dente. Drain and set aside.

Prepare the Vegetables:
In a large skillet or saucepan, melt butter over medium heat. Add diced onion, carrots, and celery. Sauté until softened, about 5-7 minutes. Add minced garlic and cook for an additional 1 minute until fragrant.

Make the Roux:
Sprinkle the flour over the cooked vegetables and stir well to coat. Cook for 1-2 minutes to eliminate the raw flour taste.

Add Liquids:
Gradually whisk in chicken broth and milk, stirring constantly to prevent lumps. Bring the mixture to a simmer and cook until it thickens, about 4-5 minutes.

Add Chicken and Peas:
Stir in cooked chicken, frozen peas, and dried thyme. Season with salt and black pepper to taste. Cook until peas are heated through.

Combine Pasta and Sauce:
Add the cooked penne pasta to the sauce, mixing until well coated.

Prepare Puff Pastry:
Transfer the pasta mixture into an oven-safe baking dish. Roll out the thawed puff pastry sheet to fit the dish if necessary. Place the puff pastry over the pasta, trimming edges if needed. Brush the pastry with the beaten egg for a shiny, golden finish.

Bake:
Preheat oven to 400°F (200°C). Bake the assembled dish for 20-25 minutes, or until the puff pastry is puffed and golden brown.

Serve:
Let cool for a few minutes before serving. Enjoy your warm, comforting Chicken Pot Pie Pasta!