Place the puff pastry onto a clean work surface and gently roll it out so that it is slightly thinner. Use the whole sheet to cut out 2 inch square pieces and store them in the fridge for later.
Preheat the oven to 400˚F. Heat a 10 inch cast iron skillet over medium high heat. Place the butter into the skillet and allow it to melt. Add in the onion, shallots, and garlic. Cook until the shallots turn translucent, about 2 minutes.
Mix in the carrots, potatoes, and celery. Cook the veggies for about 5 minutes until they've slightly softened. Pour in the white wine and cook until half of it has evaporated. Mix the flour into the vegetables and cook for 1 minute. Add in the chicken broth, poultry seasoning. Mix then let it come to a gentle bubble. Add in the shredded chicken, peas, half and half, salt, and pepper.
Remove the skillet from heat then quickly layer on the squares of puff pastry until the top is covered entirely. Brush the puff pastry with egg wash then bake it in the oven for 25-30 minutes until golden brown and bubbling. Remove the chicken pot pie from the oven and allow it to sit for 10-15 minutes before serving. Enjoy!