Brush the chicken pieces on all sides with the oil.
In a large casserole or heavy Dutch oven, combine the onions, celery, parsley and tarragon. Lay the chicken pieces over the vegetables and herbs, and pour the vermouth over the chicken. Sprinkle the chicken with salt, pepper and nutmeg. Distribute the unpeeled garlic cloves throughout the casserole, tucking them under the chicken pieces. Cover the casserole tightly (you might fit a piece of foil around the top under the lid).
Bake the chicken in a preheated 325 degree oven for 1 1/2 hours. Do not uncover the casserole until this time has elapsed.
Serve the chicken with the garlic, advising the diners to squeeze the flesh from its papery coat. The garlic is especially tasty when eaten on crusty bread.