In a small saucepan, combine the broth and garlic. Bring to a simmer over medium, then cover, reduce to medium-low and cook, turning the garlic head once or twice so it cooks evenly, until a skewer inserted into the center cloves meets no resistance, about 30 minutes. Using a slotted spoon, transfer the garlic to a plate and set aside until cool enough to handle. Pour the broth into a 2-cup liquid measuring cup or small bowl; set aside.

While the garlic simmers, heat the oven to 425°F with a rack in the middle position. Coat a 9-by-13-inch baking dish with about 1 tablespoon melted butter. Peel and slice the potatoes into ⅛-inch-thick rounds, adding them to a large bowl. Add the remaining butter, the rosemary, 1¼ teaspoons salt and ½ teaspoon pepper, then toss until the slices are evenly coated.

When the garlic is cool, squeeze the cloves out of the skins into the reserved broth; discard the skins. Using a fork, mash the cloves until mostly smooth (some chunks are fine), then stir in the mustard. You should have 1¼ cups garlic-mustard broth; if not, supplement with additional broth. Pour the mixture into the prepared baking dish.

Shingle the potatoes down the length of the baking dish, creating 3 or 4 tightly packed rows. Scrape any butter and seasonings remaining in the bowl onto the potatoes, then cover with foil. Bake until a skewer inserted into the center of the tian meets no resistance, about 45 minutes.

Remove the dish from the oven; leave the oven on. Remove the foil, then sprinkle the Parmesan evenly over the potatoes. Bake, uncovered, until the cheese is golden brown, about 30 minutes. Cool on a wire rack for 20 minutes. Serve warm.