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Preheat oven to 425°F.
Toss peeled carrots with 1 tablespoon oil, ¼ teaspoon salt and ¼ teaspoon pepper in a 9-by-13-inch metal baking pan (do not use a glass pan, which could shatter).
Spread the carrots in a single layer.
Roast until lightly browned and tender, 40 to 45 minutes, flipping halfway through.
Carefully pour ½ cup broth and 1 tablespoon lemon juice into the baking pan.
Roast until most of the liquid is absorbed, 8 to 10 minutes.
Transfer the pan to a heatproof work surface.
Immediately add 1½ tablespoons butter; gently shake the pan back and forth until the butter melts and emulsifies into the sauce.
Sprinkle with ½ teaspoon lemon zest and 2 teaspoons dill.