How to Make It
Preheat the oven to 375°. In a large roasting pan, combine the carrots, onions, potatoes, mushrooms, garlic, thyme and olive oil and toss well. Spread the vegetables in an even layer and add 4 cups of the chicken stock. Set the pan over 2 burners and bring to a boil over moderately high heat.
Meanwhile, season the chickens and duck inside and out with salt and pepper. Set the chickens, breast side up, on the vegetables. Set the duck on a rack in another roasting pan. Roast for about 2 hours, or until the birds are just cooked through and the vegetables are tender. Transfer the chickens and duck to a carving board and let cool slightly.
Discard the skin from the chickens and duck, remove the meat and cut it into 2-inch pieces. Strain the chicken pan juices, reserving the vegetables. Skim the fat from the juices.
Melt the butter in a medium saucepan. Stir in the flour and cook over moderate heat until bubbling, about 1 minute. Gradually whisk in 1 cup of the pan juices, then whisk in the remaining pan juices and 3 cups of chicken stock and bring to a boil. Simmer over moderately low heat, whisking often, until no floury taste remains, about 12 minutes. Transfer the sauce to a large saucepan and add the reserved vegetables and the chicken and duck meat.
Preheat the oven to 450°. On a lightly floured surface, roll out half of the puff pastry 1/8 inch thick. Using a 7-inch bowl, cut out 4 rounds. Repeat with the remaining pastry. Spoon the stew into eight 2- to 3-cup ovenproof bowls, about 6 inches wide. Set a puff pastry round on each bowl and press the pastry around the rim to seal. Brush the pastry with the beaten egg and arrange the bowls on a large baking sheet. Bake for about 25 minutes, or until the pastry is puffed and deeply browned. Serve at once.