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Position rack in upper third of oven; preheat to 500°F.
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Toss potatoes, butter, oil, thyme, rosemary, salt and pepper in a large bowl. Arrange in a single layer in a 9-by-13-inch metal baking pan. (Do not use a glass dish, which could shatter.) Roast, flipping once, until browned, about 30 minutes.
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Carefully add broth and garlic to the pan. Continue roasting until most of the broth is absorbed and the potatoes are very tender, about 15 minutes more. Serve hot.
Equipment
9-by-13-inch metal baking pan