Lightly grease the ramekin or baking vessel with nonstick spray. Measure the amount of puff pastry needed to line the inside. Lightly press the puff pastry in an trim any excess off. Set aside. Cut two circles from the puff pastry for the top as well.
Add the oil to a saucepan or pot over medium heat. Once hot, saute the carrots, celery, onion, and garlic until softened.
Stir in the flour until fully combined. Then stir in the broth, chicken, and thyme. Bring to a simmer to build flavor, about 10 to 15 minutes. Season with salt and pepper to taste.
Spoon the mixture into the prepared puff pastry vessels. Place a trimmed piece of puff pastry over top.
Bake for 20 to 30 minutes until golden brown.
Cool for about 5 minutes before serving. Garnish with parsley.