Preheat the oven to 400℉
In a 12-inch cast iron skillet, melt butter over medium heat. Add onion, carrot, and celery. Cook until veggies are soft and onion is translucent, about 5-6 minutes. Add minced garlic and thyme and saute until fragrant, 1 minute.
Sprinkle flour over the veggies and stir to coat. Cook and sauté, 1-2 minutes. Whisk in chicken broth, stir to avoid lumps.
Once smooth, stir in milk. Bring to a low simmer and cook until just starting to bubble and thicken, about 3-4 minutes.
Remove skillet from heat, and stir in shredded chicken, frozen peas, salt and pepper and gently fold to combine.
On a lightly floured surface, roll one sheet of puff pastry into a 16-inch square. Cut into 3-inch squares (you should have about 16-20 pieces). Arrange the squares on top of the chicken mixture, overlapping slightly. In a small bowl, beat the egg with 1 tablespoon of water. Brush the pastry with the egg wash.
Bake for 25–30 minutes, or until the puff pastry is deep golden brown and crisp.
Let rest for 10ish minutes before serving - it will be hot! Spoon directly from skillet.