Drain one block extra-firm tofu and pat dry with a kitchen towel. Cut tofu in half horizontally through the equator (like a hamburger bun), then cut into 1" cubes. Arrange tofu cubes in a single layer on a rimmed baking sheet or large plate and set aside.
Cut broccoli into small 1" florets. If your broccoli has a long stem, peel tough outer skin to remove the lighter fibrous layer, then slice stem ¼" thick.
To make the sauce, scrape skin from one 2" piece ginger with a spoon, then grate on a Microplane into a small bowl. Peel garlic and grate into bowl with ginger. Add tamari or soy sauce, brown sugar or pure maple syrup, tahini, rice vinegar, toasted sesame oil, 3 tsp. cornstarch, a pinch of kosher salt, and 6 Tbsp. water and whisk to combine. Set sauce aside.
Sprinkle 1 tsp. salt over tofu and toss to coat. Scatter approximately ¼ cup cornstarch over tofu, a tablespoonful at a time, tossing after each addition and gently pressing into the tofu pieces, until well coated on all sides.
Warm 1 Tbsp. grapeseed oil or vegetable oil in a large nonstick or cast-iron skillet over medium-high heat. Add broccoli, season with ½ tsp kosher salt and ¼ tsp. freshly ground black pepper, and cook, tossing occasionally, until just tender, about 5 minutes. Transfer broccoli to a plate. Wipe out skillet. (I roasted the broccoli in the oven at 375F, as opposed to sauteing in pan)
Heat remaining ¼ cup grapeseed or vegetable oil in same skillet over medium-high. When oil is hot (it will start to shimmer), reduce heat to medium and add tofu in a single layer (work in batches if needed). Cook, turning every 3–4 minutes, until golden brown all over, 10–12 minutes total. Transfer crispy tofu to a clean kitchen towel or paper towels and let drain. (I roasted the tofu in the oven at 375F, as opposed to sauteing in pan).
Pour out any remaining oil in pan and wipe out. Return pan to medium heat. Whisk reserved sauce if it has separated, then pour into pan and cook until thickened and bubbling, about 20 seconds. If it looks like you have too much sauce, after thickening pour some in a small dish and serve alongside the meal.
Immediately remove pan from heat and add broccoli and tofu; toss to coat.
Garnish tofu and broccoli with toasted sesame seeds and sliced scallions. Serve with steamed white rice or brown rice.