Bring a large pot of salted water to a boil. Add 8 ounces of penne pasta and cook according to the package instructions until al dente. Drain and set aside.
Heat 2 tablespoons of olive oil in a large skillet over medium heat. Add the diced onion, carrots, and celery. Cook for about 5-7 minutes, stirring occasionally, until the veggies are softened. Toss in the minced garlic and cook for another minute until fragrant.
Pour in 3 cups of chicken broth and bring to a simmer. Stir in 1 cup of heavy cream, 1 teaspoon of dried thyme, and 1 teaspoon of dried rosemary. Let the sauce reduce slightly, about 5 minutes, until it thickens and coats the back of a spoon. Season with salt and pepper to taste.
Add the shredded chicken and frozen peas to the skillet, stirring gently to combine. Then fold in the cooked penne pasta, making sure every piece is coated in that luscious sauce. Remove from heat and stir in 1 cup of shredded cheddar cheese until melted and gooey.
Transfer the entire mixture to a greased baking dish. Roll out the thawed puff pastry to fit over the top of the dish, trimming any excess. Lay the puff pastry gently over the pasta mixture and press the edges to seal. Cut a few small slits in the pastry to allow steam to escape during baking.
Preheat your oven to 400°F (200°C). Bake the pasta bake for 25-30 minutes or until the puff pastry is puffed up and golden brown. Let it rest for 5 minutes before serving to allow the filling to settle.