Preheat your oven to 375°F (190°C). Grease a 9×13-inch baking dish.
Cook the penne pasta according to the package instructions until al dente. Drain and set aside.
In a large skillet, melt unsalted butter over medium heat. Add diced onion, carrots, and celery. Cook until vegetables are softened, about 5-7 minutes.
Add minced garlic and cook for an additional 1-2 minutes, until fragrant.
Sprinkle all-purpose flour over the vegetables and stir to combine. Cook for 1-2 minutes to remove the raw flour taste.
Gradually whisk in chicken broth and milk until smooth. Bring the mixture to a simmer, stirring frequently, until it thickens.
Stir in diced chicken, frozen peas, dried thyme, salt, and black pepper. Cook for another 2-3 minutes.
Add the cooked penne pasta to the skillet and stir until well coated with the sauce.
Transfer the chicken pot pie pasta mixture to the prepared baking dish, spreading it evenly.
Roll out the puff pastry sheet to fit the size of the baking dish. Place it over the chicken pot pie pasta mixture.
Brush the puff pastry with beaten egg for a golden finish.
Bake in the preheated oven for 25-30 minutes, or until the puff pastry is golden brown and the filling is bubbly.
Remove from the oven and let it cool for a few minutes before serving.