Rinse the chicken in cool water, then pat dry with clean paper towels
Set a heavy-bottomed frying pan over moderately-high heat
Add 2 tablespoons of butter and 1 tablespoon of olive oil to pan
When very hot (but not smoking) place the chicken pieces skin-side down and season with salt and pepper
Saute for 5-6 minutes, basting with the juices and fat, until underside is browned.
Turn pieces over, season again with salt and pepper, and saute the same way for an additional 5-6 minutes, or until juices run clear.
While the chicken is cooking, in a separate pan saute the mushrooms in 1 tablespoon butter, remove to a plate when finished browning
When chicken is finished cooking, remove to a warm plate
Rapidly spoon all but a tablespoon of fat out of the pan.
Add the 1/2 cup of wine to deglaze the pan, scraping the sides and bottom to incorporate all the seasoning into the liquid. Deglaze until the alcohol has burned off
Add cream and mushrooms to the deglazing sauce and incorporate; boil down rapidly for a minute or two, or until the sauce starts to thicken
Add the chicken and baste with the sauce and mushrooms.
Cover and simmer 2-3 minutes to bring chicken back up to a hot temperature
Season again if necessary with salt and pepper