Heat the oven to 220C (425F). Press the tofu between paper towels under something heavy for 10 minutes.
Tear the tofu into rough bite-sized chunks with your hands. Do not cut it. The ragged torn surfaces are what go crispy and grab the sauce; knife-cut cubes stay smooth and slick.
Toss the chunks with the oil, salt, and 3 tbsp cornstarch until coated and a little shaggy. Spread on a lined baking sheet with space between pieces.
Bake 25 to 30 minutes, flipping once halfway, until deep golden and crackly at the edges. An air fryer does it in about 15 minutes at 200C.
Meanwhile, simmer the soy sauce, mirin, sugar, and ginger in a small pan for 2 minutes. Stir in the cornstarch slurry and cook until it turns glossy, about 30 seconds. Take it off the heat.
Cook the edamame in boiling salted water for 3 minutes and drain.
Toss the hot tofu in the glaze just before serving, not earlier. Pile over rice with the edamame, then finish with scallions and sesame seeds.