Press tofu for 45 minutes.

Meanwhile, preheat oven to 425°. Cut broccoli heads into florets. Thinly slice scallions and separate green and white parts. Into a medium bowl or large measuring cup, grate ginger and garlic. Add white scallion parts, broth, soy sauce, maple syrup, chili crisp, vinegar, sesame oil, and 1 tablespoon cornstarch and whisk to combine.

Break tofu into rough 1" pieces and transfer to a large bowl. Sprinkle with ½ teaspoon salt, drizzle with 1 tablespoon neutral oil, and gently toss tofu to coat. Sprinkle with remaining ⅓ cup cornstarch and gently toss until tofu is coated. Arrange on a parchment-lined baking sheet and drizzle with 2 tablespoons neutral oil.

Bake tofu, turning halfway through, until light brown and crisp, about 25 minutes.

When tofu has about 12 minutes remaining, in a large skillet or wok over medium-high heat, heat remaining 1 tablespoon neutral oil. Add broccoli; season with salt, then toss to coat broccoli with oil. Cook, stirring frequently, until broccoli is starting to char, 4 to 5 minutes. Transfer broccoli to a plate.

Reduce heat to medium. Whisk sauce to reincorporate cornstarch and pour into skillet. Bring to a simmer and cook, stirring occasionally, until slightly thickened, about 4 minutes. Add broccoli and tofu to skillet and toss to coat in sauce. Top with green scallion parts and sesame seeds. Serve over rice.