Cut the tofu into slabs, 1 cm thick. Pat dry with paper towels, then cut into 1-cm cubes. Heat olive oil in a 10-inch skillet over medium heat. Once hot, add half of the cubes of tofu. They should sizzle immediately. If the pan spits water excessively, it's either too hot or you didn't dry the tofu thoroughly enough. Fry, tossing occasionally, until evenly browned (about 15 minutes). Fry the second batch in the same way, adding more oil if needed.

While the tofu cooks, mince the garlic and ginger and slice the scallions. Whisk together the ingredients for the sauce. Once the tofu is done, remove from the pan to a plate lined with paper towels. Wipe the broken bits of tofu and most of the oil from the pan, then add the garlic, ginger, and scallions. Cook briefly, taking care not to burn the garlic. Add the sauce and stir to combine. It should thicken substantially but maintain a sauce-like consistency. If it's too thick, add a little water.

Add the tofu back to the pan and stir, briefly, to coat. Sprinkle in the scallions and sesame seeds to coat the tofu. Serve with rice and roasted broccoli.