How to Make It
In a bowl, blend together the flour, butter and 1/2 teaspoon salt with your fingertips or a pastry blender until the butter is mostly combined with some pea-sized lumps. Using a fork, stir in 3 tablespoons of the water. Squeeze a handful of the dough, if it is still crumbly, stir in the remaining 1 tablespoon water. Gather the dough into a ball, then form it into a disk and wrap it in plastic wrap. Chill the dough at least 1 hour.
On a floured work surface with a floured rolling pin. roll the dough out into a 10-inch round. Using a round cutter, cut 4 3-inch rounds from the dough. Chill the rounds until ready to use. Preheat the oven to 400°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, 1/2 teaspoon salt and 1/4 teaspoon pep-per. 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. Divide the filling between 4 (8-ounce) ramekins. Place the pastry rounds over the ramekins and brush with some of the egg mixture. Cut a steam vent in each pastry round. Bake the individual pot pies until the filling is bubbling and the pastry is golden, 20 to 25 minutes.