Prepare the morels: Place the morels in a colander and rinse well under cold running water to rid them of any grit. Transfer them to a heatproof measuring cup. Pour boiling water over the mushrooms to cover. Set aside for 20 minutes to plump them up. Then, using a slotted spoon, carefully remove the mushrooms from the liquid, leaving behind any grit that may have fallen to the bottom. If any of the morels are extremely large, halve them lengthwise.
Liberally season the chicken pieces on all sides with sea salt and white pepper.
In a deep 12-inch skillet, combine the oil and 4 tablespoons of the butter. Place over moderate heat. When the fats are hot but not smoking, add the chicken, skin side down, and brown until it turns an even golden color, about 5 minutes more. Carefully regulate the heat to avoid scorching the skin. (This may have to be done in batches.) When all the pieces are browned, use tongs (to avoid piercing the meat) to transfer them to a platter. Season again with sea salt and white pepper.
Pour off and discard the fat in the skillet. Add the remaining 1 tablespoon butter, the shallots, and the garlic. Cook, covered, over low heat, until soft but not browned, about 3 minutes. Add the wine and boil, uncovered, over high heat for 5 minutes to burn off the alcohol, which could make the sauce bitter. Add the cream and the morels, and stir to blend. Return the chicken and any juices that have accumulated to the skillet. Cover, and cook over low heat, turning the pieces in the sauce once or twice, until the chicken is cooked through and has thoroughly absorbed the sauce, about 20 minutes. Taste for seasoning and serve.