Cabbage Rolls with Meat Stuffing and Wild Mushroom Sauce

2 large heads savoy cabbage 5 tbsp/70 g unsalted butter 1/4 cup/40 g coarsely chopped onion 4 cups/630 g cooked white rice 1 lb/455 g coarsely chopped boneless pork or chicken (thigh meat will be juicier, but white meat works as well) Salt and freshly ground pepper 2 cups/480 ml chicken broth or stock