Remove 1 puff pastry sheet from the freezer and let it begin to thaw.
Preheat the oven to 400 degrees. Line a baking sheet with parchment paper.
Cut 1 sheet of puff pastry into 6 pieces and set them on the baking sheet to continue to thaw.
Place the butter in a large skillet on the stove over medium heat.
Once the butter melts, add the minced garlic, onion, carrots, and celery. Let the mixture cook for 3-4 minutes or until the vegetables have softened.
Add the salt, thyme, and black pepper and mix well.
Add the flour and stir until the vegetables are evenly coated in flour.
Add the milk and chicken broth and stir continuously until the mixture begins to thicken, about 2-3 minutes.
Once the mixture has thickened, add the peas and chicken to the skillet and mix until well combined.
Turn the heat to low and let the mixture simmer for 5 minutes, stirring frequently to prevent burning.
Remove the skillet from the heat.
Crack the egg in a small bowl and whisk it.
Use a pastry brush to brush a light layer of egg over the tops of the puff pastry.
Place the baking sheet in the oven and let the puff pastry bake for 8-10 minutes, or until the tops are golden brown.
Serve the chicken pot pie mixture in a bowl with a piece of puff pastry on top.
