Place the chicken in a medium saucepan and cover with 2 1/2 cups of chicken broth. Add a bay leaf, garlic, and a few sprigs of fresh herbs, if desired. Bring to a boil; cover and reduce the heat to low. Simmer for 15 minutes until the chicken breasts reach at least 165°F. Strain the broth and set it aside. Shred or chop the chicken breasts and set aside.
1 pound chicken breasts | 2 1/2 cups chicken broth | 1 bay leaf | 3 cloves garlic | A few sprigs of fresh herbs (optional)
For the Chicken Pot Pie
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.
1 tbsp olive oil | 1 tbsp unsalted butter | 1 cup diced onion | 1 cup diced celery
Add the potatoes and 1 3/4 cups of the reserved chicken broth to the skillet and bring to a boil. Reduce the heat to low and simmer for about 15 minutes, or until the potatoes are tender.
12 to 16 ounces potatoes
Add the chicken, peas, and carrots to the skillet and simmer for 2 minutes.
1 1/2 cups frozen peas and carrots
Combine the heavy cream and cornstarch in a cup or small bowl and whisk to make a smooth slurry. Add to the skillet and cook, stirring, until thickened, adding more of the reserved broth as needed.
2/3 cup heavy cream | 2 tbsp cornstarch
Taste and season with kosher salt and freshly ground black pepper, as needed.
Kosher salt and freshly ground black pepper
Transfer the filling to a 1 1/2- to 2-quart baking dish, oven-safe skillet, or large pie plate and let it cool slightly.
Let the filling cool slightly before topping with the pastry so the puff pastry stays cold and bakes up crisp and flaky.
Position a rack in the lower third of the oven and heat the oven to 400°F.
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.
1 sheet puff pastry
Whisk the egg in a small bowl with 1 tablespoon of water. Brush lightly over the patches.
1 large egg | 1 tbsp water
Bake the pie for 25 to 30 minutes or until the topping is golden brown and the filling is bubbling. If the topping is getting too dark, tent it with a sheet of foil.
Serve with a garnish of fresh chopped parsley if desired.