Preheat oven to 350°F (Gas Mark 4). Grease a 9x13-in baking dish with butter and set aside.
Heat milk with garlic in a medium saucepan over medium heat until small bubbles rise to surface. Remove and discard garlic. Add 1 cup cheese and whisk until melted.
Beat egg in a large bowl and gradually add cheese mixture, then salt, black pepper and nutmeg. Whisk Chobani into mixture in three additions.
Place sliced potatoes in two or three overlapping rows in prepared baking dish. Pour yogurt-cheese mixture over potatoes and sprinkle with remaining cheese. Bake until golden-brown and potatoes are tender when pierced with a paring knife, 40-50 minutes. Turn broiler on and broil gratin until golden-brown on top, 2-3 minutes. Let sit 10 minutes before serving.