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Preheat oven to 425°F.
Place peeled garlic cloves on a piece of foil big enough to enclose them. Drizzle with 1 teaspoon oil; seal the foil securely. Place in a 9-by-13-inch metal baking pan and roast for 10 minutes (do not use a glass baking pan, which could shatter when the broth is added later).
Meanwhile, toss broccoli, ¼ teaspoon each salt and pepper and the remaining 2 tablespoons oil together in a large bowl. After the garlic has roasted for 10 minutes, move it to one side of the pan; arrange the broccoli in a single layer on the other side. Roast until the broccoli is browned in spots and slightly tender, about 15 minutes.
Remove the pan from the oven; open the garlic packet. Using a fork, mash the garlic together with the oil until a paste forms. Whisk ⅓ cup broth, 1 tablespoon lemon juice and the remaining pinch of salt together in a measuring cup; add the mashed garlic and whisk again. Carefully add the broth mixture to the pan with the broccoli. Roast until the liquid is mostly reduced, 7 to 10 minutes more. Sprinkled with 1 tablespoon Parmesan.