Preheat oven to 450°. Remove chard stems and discard; chop leaves to yield about 3 cups. Thinly slice potatoes (about ⅛ inch thick), using a mandoline or food processor if possible, and set aside.

Pour half-and-half or milk into a 2-quart casserole dish. Layer mushrooms on top, then add a layer of chard. Sprinkle garlic over chard. Arrange sliced potatoes on top of garlic and chard. Sprinkle half of fontina or Muenster cheese on top of potatoes. Cover with foil and bake until potatoes are tender, 25-30 minutes.

Preheat broiler. Remove foil; sprinkle gratin with remaining fontina or Muenster cheese and blue cheese. Broil until cheese melts, about 2 minutes. For a firmer texture, let stand for 5 minutes before serving.