1. Preheat the oven to 350°F.

2. Heat the butter or oil in a medium saucepan over medium-high heat until it shimmers. Add the onion and sauté, stirring frequently until tender, 10 to 12 minutes. Add the garlic and sauté until aromatic, about 30 seconds. Add the flour and cook, stirring constantly, until pasty and thick, about 2 minutes. Add the broth, whisking well to work out any lumps. Bring to a boil and them immediately reduce the heat to low and simmer, stirring frequently, until thick, about 15 minutes. Season to taste with salt and pepper.

3. Add the carrot, celery, and potato and simmer until the vegetables are tender, about 20 minutes. Add the chicken and peas and remove from the heat. Season to taste with salt and pepper. Stir in the chopped parsley.

4. Spoon the filling into individual crocks or a baking dish. Cut pie crust or puff pastry dough to the approximate size and shape and cover the filling. Cut vents in the crust and press the edges of the dough onto the baking dish or crocks to seal.

5. Bake the pot pie until the pie crust or puff pastry is golden and flaky, about 45 minutes for a large pot pie and 25 minutes for individual crocks. Serve immediately.