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Preheat the oven to 400.
Heat the olive oil in an oven-safe, heavy-bottomed 12-inch skillet over medium heat until shimmering. Add the onion, celery, and garlic, and sprinkle with a pinch of kosher salt. Cook for about 5 minutes, until the vegetables are softened but not browning.
Stir in the carrots, green beans and potatoes, then add the chicken broth. You may not need all 3 cups of broth; only add as much as you need to barely cover the vegetables. Bring to a simmer and cook for about 10 minutes until the broth is slightly reduced and absorbed, then taste for seasoning and add salt as needed.
Sprinkle the flour evenly over the vegetables and stir into the mixture. Add the chicken and bring the liquid back to a simmer.
Unfold the sheet of puff pastry and gently use a rolling pin if needed to stretch it to the width of the skillet.
Whisk the egg with 1 tablespoon water to make an egg wash.
Remove the skillet from the heat and place the puff pastry over the vegetables and sauce, tucking the edges of the pastry into the skillet to completely cover the filling. Brush the pastry with the egg wash.
Bake for 15 minutes, until the pastry is puffed and golden. Let the pot pie rest for 10 minutes before serving.