Whisk all of the sauce ingredients together in a small bowl. Set it aside.
Place the cornstarch in a small bowl.
Generously coat the bottom of a large skillet with oil and place it over medium heat. You'll need the oil to be about ¼ inch deep.
While the oil heats up, place a few tofu cubes into the bowl of cornstarch. Dredge the tofu in the cornstarch, then once the oil begins to shimmer, remove the tofu from the bowl and gently shake the pieces to remove any excess cornstarch. Add the tofu cubes to the hot oil. Continue coating the tofu and adding pieces to the oil, cooking as many at a time as you can fit without crowding.
Cook the tofu for about 5 minutes, until lightly browned and crispy, then flip the pieces and cook them for another 5 minutes on the opposite sides. Transfer the cooked tofu to a plate when done, and cook any remaining pieces in a second batch, adding oil to the skillet if needed.
If the skillet is dry at this point, add a dash more oil. If there's already more than a light coating of oil, carefully blot some out with a paper towel. Raise the heat to high.
Add the dried chiles to the skillet, then stir-fry them for about 1 minute, until they begin to darken.
Add the broccoli to the skillet and stir-fry until tender-crisp and bright green, about 3 minutes.
Push the broccoli and dried chiles to the sides of the skillet, then add the white parts of your scallions, garlic, and ginger to the center of the skillet. Sauté the mixture for about 30 seconds, stirring constantly, until very fragrant.
Return the tofu to the skillet, then carefully pour the sauce over the tofu. Flip everything a few times to coat the broccoli and tofu with the sauce. Cook everything just until the sauce thickens, about 1 minute.
Remove the skillet from the heat and sprinkle with green parts of scallions and sesame seeds.
Divide onto plates and serve with rice.