Carefully pat tofu dry: On a cutting board, sandwich tofu between doubled layers of paper towels and press down slowly to remove moisture without cracking the block. Remove paper towels and cut tofu into 32 equal pieces: halve tofu laterally across, then cut the block into a 4×4 grid.
Make marinade: In a medium bowl, whisk together garlic, ginger, soy sauce, BBQ sauce, 2 tablespoons sesame oil, vinegar, and sugar until smooth. Set aside half this mixture and reserve for glazing later.
Preheat the oven to 400°. Meanwhile, add tofu to the bowl and toss gently to coat in marinade. Let sit for 15 minutes, then drain off excess marinade.
In a small bowl, whisk together cornstarch and baking powder. Sprinkle over tofu and gently toss to coat evenly. Drizzle over remaining 1 tablespoon sesame oil and toss to coat each piece.
Spread tofu evenly out onto a baking sheet and bake for 15 minutes. After 15 minutes, decrease oven temperature to 375°. Flip tofu pieces to allow for even toasting, then return the tray to the oven and continue to bake until tofu is crispy on the outside and deeply golden, about 15 minutes more.
Brush with reserved marinade mixture for a glaze finish, then sprinkle with green onions and sesame.