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Position rack in upper third of oven; preheat to 500°F.
Peel and discard papery outer layers of garlic head. Cut about ¼ inch from the top of the cloves. Place the garlic on a piece of foil big enough to enclose the head.
In a large bowl, combine 2 tablespoons oil, 1 tablespoon melted butter, 1 tablespoon sage, 1 teaspoon cumin and ½ teaspoon each salt and pepper. Drizzle 1 teaspoon of the oil mixture over the garlic head. Seal the foil securely.
Add 6 cups sliced sweet potatoes to the remaining oil mixture and toss to coat. Arrange in a single layer in a 9-by-13-inch metal baking pan. (Don’t use a glass pan, as it could shatter.) Place the foil-wrapped garlic in the pan; roast for 20 minutes.
Remove the garlic and set aside. Flip the sweet potatoes; roast until well browned on both sides, 7 to 10 minutes more.
Meanwhile, when cool enough to handle, squeeze the garlic cloves into a measuring cup. Add 1 cup broth and whisk to combine. When the potatoes are browned, carefully add the broth to the pan. Roast until most of the liquid has been absorbed and the potatoes are very tender, about 15 minutes.