In a large saucepan over high heat, bring 8 cups (64 fl oz/2 l) water and 1½ teaspoons salt to a boil. Add the polenta in a slow, steady stream, stirring constantly. Reduce the heat to low and cook, stirring often, until the polenta pulls away from the sides of the pan, 40–45 minutes.
Meanwhile, prepare the chard and mushrooms. Bring a large pot of water to a boil over high heat. Add the chard leaves, folding them to fit, and the rosemary sprig. Reduce the heat to medium and cook until the chard ribs are easily pierced with a fork, about 15 minutes. Drain well. Chop coarsely and squeeze dry. Set aside.
In a frying pan over medium-high heat, warm the olive oil and 1 tablespoon of the butter. Add the shallots and mushrooms and cook until the mushrooms are tender, 8–10 minutes. Using a slotted spoon, transfer to a bowl. Reserve the juices.
When the polenta is ready, stir in the remaining 3 tablespoons butter, all but ¼ cup (1 oz/30 g) of the cheese, 1 teaspoon salt, and 1 teaspoon pepper and cook until the butter and cheese have melted, 3–4 minutes longer. Return the frying pan you cooked the mushrooms in to medium-high heat. Warm the juices, then add the chard and mushrooms and cook, stirring, until hot and well coated with the juices. Season with ½ teaspoons each of salt and pepper.
Spoon the polenta into a large serving bowl, top with the chard and mushrooms, and sprinkle with the remaining cheese. Serve at once.