Preheat the oven to 350 degrees.
In a 3-quart mixing bowl, gently mix the potatoes, ham and mushrooms. In a 2-cup saucepan, combine the mushroom soup, heavy cream, chicken broth, tarragon, oregano, and salt and pepper. Warm the mixture at medium heat, reduce the heat to low and simmer for 5 minutes, stirring occasionally.
Spread the potato-ham-mushroom mixture evenly in a greased 3-quart baking dish. Pour the mushroom soup sauce over this. If you need a bit more sauce, add whole milk until the potatoes are just covered.
Bake, uncovered, for 1-1/2 to 1-3/4 hours or until the potatoes are tender when pierced with a fork. Sprinkle the sliced onion tops over the dish during the last 10 to 15 minutes of baking for a garnish. Remove from the oven and let rest for 10 minutes before serving. Serves 4.