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In a large cast-iron skillet, melt the butter over medium-low heat. Add the onions, celery, garlic, carrots, and thyme, and season with 1 teaspoon each salt and pepper. Sweat the mixture until the vegetables are soft and translucent, about 10 minutes. Reduce the heat to low, add the flour, and stir for 2 minutes.
Add the chicken stock to the sauce, turn the heat to medium, and continue stirring until the mixture has thickened. Add in the half-and-half, and fold in the shredded chicken, peas, and fresh parsley. Season to taste with salt and pepper.
Preheat the oven to 375°F.
On a floured work surface, roll out the chilled pie crust until its diameter is about 1-inch bigger than the diameter of your cast-iron skillet. Carefully lay the dough over the filling and cut slits in the top to create steam vents. Gently crimp the edges to seal, lightly brush the crust with the egg wash, and sprinkle with the flaky salt.
Bake until the filling is bubbly and the crust is golden brown, about 35-40 minutes. Allow the pot pie to rest for 5 minutes before slicing and serving.