Wrap tofu in a clean towel and press under a moderately heavy plate for 25 minutes. Heat the oven to 425°F and line an 18-by-13-inch rimmed sheet pan with parchment rated for that temperature.
Cut tofu into 1-inch cubes and pat dry. Toss with 1 tablespoon oil, cornstarch, paprika and 1/4 teaspoon salt and spread over one half of the pan.
Roast tofu for 15 minutes. Toss broccoli and onion with remaining oil and salt and the pepper, add to the other half of the pan and turn the tofu cubes.
Roast for 15 minutes more until tofu corners are firm and broccoli edges brown. Mix barbecue sauce and cider vinegar in a small bowl.
Coat tofu lightly with the sauce and roast for 5 to 7 minutes more until tacky. Divide tofu and vegetables among 3 plates.