Preheat oven to 400°F.
Divide the butter into three pieces. Rub one-third of the butter all over the chicken. Season with salt, pepper and fennel. Place the garlic, thyme, lemon and bay leaf into the cavity of the chicken. Place the chicken breast side up in a roasting pan and roast for 50 to 60 minutes or until a thermometer in the leg registers 160°F, basting a few times throughout.
Let the chicken rest until it is cool to the touch. Reserve the juices. Remove all white and dark meat, reserving the skin and bones for stock.
Heat a Dutch oven over moderate heat and add one-third of the butter plus the olive oil, vegetables, tarragon, and parsley. Cover and stir often with a wooden spoon until the vegetables are fork tender, about 10 minutes. Add the remaining butter, the flour and the reserved juices. Stir until well mixed. Add the stock, cream and reserved chicken meat and cook for 10 minutes over medium-high heat until the sauce is creamy. Turn off the heat. Add the peas.
Place the filling into four 12-ounce ceramic dishes and top each with a piece of puff pastry to cover the entire dish and drape down a little on all sides. Cut two slashes in the top and brush with a beaten egg.
Place the dishes on a sheet tray and put them in a 400°F oven for about 15 minutes, until the dough is golden.
Makes 4 individual pot pies