Place the chicken in a medium saucepan and cover with 2 cups of chicken stock. Add a bay leaf, garlic, and some fresh herbs. Bring to a boil; cover and reduce the heat to low. Simmer for 15 minutes until the chicken breasts reach at least 155 F. Strain the broth and set it aside. Shred or chop the chicken breasts and set aside.

Heat the olive oil and butter in a large skillet over medium heat. When the butter begins to sizzle, add the onion and celery; cook for 4 to 6 minutes, until softened.

Add the potatoes and 1 3/4 cups of the reserved chicken cooking broth to the pot and bring to a boil. Reduce the heat to low and simmer for 15 minutes, until the potatoes are tender.

Add the chicken and peas and carrots to the pot and simmer for 2 minutes.

Combine the heavy cream and cornstarch in a cup or small bowl and whisk to make a smooth slurry. Add to the pot and cook, stirring, until thickened, adding more of the reserved broth, if necessary.

Taste and season with kosher salt and freshly ground black pepper, as needed.

Transfer the filling to a 1 1/2 to 2-quart baking dish, oven-safe skillet, or large pie plate—let it cool to room temperature.

Position a rack in the lower third of the oven and heat the to 425.

Roll out the puff pastry into a 12-inch square and cut into 2 1/2 to 3-inch squares (about 16 squares). Arrange the squares on the filling, overlapping, leaving a few small bare spots for venting.

Whisk the egg in a small bowl with 1 tablespoon of water. Brush lightly over the patches.

Bake the pie until the topping is golden brown and the filling is bubbling.

Serve with a garnish of fresh chopped parsley, thyme leaves, or chives, if desired.