This just might be the ultimate deep-dish chicken pot pie. Lots of veggies and fresh herbs go into the tasty, creamy filling, and an impressive lattice crust makes the pie as picture-perfect as it is delicious.
Make the filling: In a large pot, heat 2 tablespoons butter over medium heat. Add the onions, celery and carrots and cook, stirring, until the onions are translucent, 5 to 6 minutes. Add the garlic and cook until fragrant, about 1 minute more.
Add the remaining 3 tablespoons butter. Once melted, stir in the flour. Cook the mixture, stirring constantly, until it just starts to turn golden brown, 2 to 3 minutes. Gradually whisk in the chicken broth. Bring the mixture to a simmer. Add the bay leaf, rosemary and thyme. Adjust the heat and simmer for 15 minutes.
Stir in the cream, chicken and peas and return to a simmer. Simmer for 4 to 5 minutes more. Remove the mixture from the heat. Stir in the salt and pepper. Ideally, cool it completely. Remove and discard the bay leaf, and stir in the parsley. Taste and add more salt and pepper, if desired.
Meanwhile, prepare the crust: Have a 10-inch cast-iron or ovenproof skillet ready. On a lightly floured surface, roll out half of the pie crust to a thickness of about 1/4 inch, making a circle about 3 inches wider all around than the skillet. Gently transfer the dough to the skillet, lining the bottom and easing it up the sides, allowing any excess to drape over the edge of the pan (aim for about 1 inch excess dough all the way around the pan). Refrigerate for 30 minutes.
While the bottom crust is chilling, roll out the remaining dough on a lightly floured surface to 1/4 inch thick. Gently transfer the dough to a parchment-lined baking sheet, then use a pastry wheel to cut it into 1 1/2- to 2-inch wide strips. Cover with plastic wrap and refrigerate for 15 minutes.
Pour the cooled filling into the chilled bottom crust and spread evenly. Remove the chilled dough strips from the fridge. Lay strips over the filling, leaving almost no space between them. When the pie is covered, rotate the pan and pull back every other strip of dough, then weave in a fresh strip to create a lattice. Lay the strips back over the new piece of dough, the repeat the process – this time pulling back all the pieces of dough that didn’t get moved the first time. Continue until the entire pie is latticed.
Press the dough together at the edges all the way around the pie; use scissors to trim away any excess dough, then fold the bottom crust up and over the lattice. Dip your finger in water and run it along the edges to help seal it. Use a floured fork to press around the edges to seal the dough together and crimp the crust. Refrigerate for 30 minutes.
Preheat the oven to 425 degrees F and place a rack in the lower third of the oven. Brush the egg mixture over the top of the pie and sprinkle it with Parmesan. Bake for 30 minutes; lower the oven temperature to 375 degrees F and continue baking until the crust is deeply golden brown (the filling may bubble through the top crust a bit), 40 to 50 minutes more.