Drain the 14 oz extra-firm tofu in a colander, then slice into roughly 1-inch-thick slabs; press between a double layer of paper towels to remove excess water, replacing towels if they become soggy, then cut the pressed slabs into 1-inch cubes.
Slice fresh ginger and peel whole garlic cloves to use for seasoning the oil; separate the white and dark green parts of the green onions and cut into 2–3 inch pieces.
Combine 1/4 cup vegetable stock with 1 tablespoon minced garlic and 1 tablespoon minced ginger in a small bowl; measure 2 tablespoons soy sauce and set aside.
Heat a wok or large frying pan over medium-high heat until hot to the hand, then add 2 tablespoons peanut oil and heat until shimmering.
Add the sliced ginger and the whole garlic cloves to the hot oil and cook, stirring, for 30–45 seconds until fragrant, then remove the ginger and garlic pieces immediately to avoid burning.
Add the tofu cubes to the pan and stir-fry, turning often, until the surfaces begin to brown, about 5 minutes.
Add the white parts of the green onions and stir-fry with the tofu until the onion whites start to soften and the tofu browns more, about 2–3 minutes.
Pour in the vegetable stock mixture (with the minced garlic and ginger) and cook, stirring occasionally, until most of the liquid has evaporated, about 2–3 minutes.
Add the dark green parts of the green onions and cook for about 1 minute, stirring a few times.
Stir in 2 tablespoons soy sauce and cook for about 1 more minute, stirring to coat the tofu evenly.
Transfer the cooked tofu to a serving dish, sprinkle with 1 teaspoon sesame seeds, and serve hot.