Preheat the oven to 425°F. In a large cast-iron or heavy skillet, heat the oil over medium high heat until hot, then stir in the onion, garlic, carrots, potato, 3/4 teaspoon salt and 1/2 teaspoon pepper. Cook, stirring occasionally, until golden, 8 to 10 minutes. Stir in the flour and cook, until incorporated, about 2 minutes. Pour in the stock and bring to a boil, then stir in the chicken. Season the filling with salt and pepper to taste.
Place the filling in a 2 1/2-to 3-quart soufflé dish. Trim the puff pastry into an 8-inch square and place it over the soufflé dish. Brush the puff pastry with some of the egg mixture, then cut a steam vent in the pastry. Bake until the filling is bubbling and the pastry is golden, about 30 minutes.
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