Heat the oven to 425°F (220°C). Line a large sheet pan with parchment.
Wrap the tofu in a clean kitchen towel and set something heavy on top, such as a skillet with a few cans in it. Press for 15 minutes.
Cut the tofu into 3/4-inch cubes. Toss them gently with 1 tablespoon of oil, then sprinkle with the cornstarch and toss again until no dry powder remains.
Spread the tofu on one half of the pan. Toss the broccoli with the remaining oil and the salt and spread it on the other half, keeping everything in a single layer.
Bake for 25–30 minutes, turning the tofu and stirring the broccoli halfway, until the tofu is golden and crisp at the edges and the broccoli is browned in spots.
Meanwhile, whisk the soy sauce, maple syrup, vinegar, sesame oil, ginger and garlic in a large bowl.
Tip the hot tofu into the sauce and toss to coat. Serve over the rice with the broccoli, and scatter with sesame seeds and green onions.