Cut the chicken up into serving pieces, making sure that the leg is separated from the thigh, and that the breast is cut in half crosswise.
Melt the butter is a braising pan large enough to contain all the chicken pieces in a single layer. Add the pieces and whole shallot, and sauté over moderate heat until the chicken is very lightly browned, salting as you go. Be careful to regulate the heat so as not to allow the butter or chicken to darken too much.
Add cream, which should cover the chicken by about 2/3. Cover (leaving the lid slightly 'ajar' to allow for evaporation and avoid spillage) and let the chicken simmer in the cream for 30-45 minutes, or until the chicken is fully cooked through and the cream reduced to a thick, saucy consistency. The cream will probably have separated, which is perfectly fine. Taste and adjust for seasoning. You can remove the shallot, which by now will be very soft or, if you like, just mash up it into the sauce.
Serve immediately. If you like, you can top the chicken with some chopped parsley, or add it to the sauce just before serving.