Preheat a large skillet and melt the ghee (or cooking fat of choice). Add the thinly sliced onions and sauté for 10-15 minutes over low heat, allowing the onions to caramelize slightly. Do not let them burn.
Move the onions to one side of the skillet, and add the halved chicken hearts and sliced mushrooms. Sprinkle the salt, pepper, and thyme around the skillet. Turn the heat up to medium-high and allow for the chicken hearts and mushrooms on the bottom to cook undisturbed for 2-3 minutes to let them brown a little. They might release juices, and that's OK.
Stir everything around in the skillet and allow to cook for 5-8 more minutes, or until the chicken hearts and fully cooked and no longer pinkish inside, stirring frequently. The mushrooms will release some juices—those are delicious over mashed potatoes or rice!!
Sprinkle with fresh herbs, stir, and serve hot with your favorite sides.