Press the tofu between paper towels or a clean kitchen towel with something heavy on top for at least 15 minutes. Cut into 1-inch cubes and toss with 1.5 tsp cornstarch until lightly and evenly coated.
Heat 0.5 tbsp of the oil in a wok or large nonstick pan over medium-high heat. Pour in the whisked eggs, let sit undisturbed for 10 seconds, then push and fold gently until just set but still a little glossy, about 45 seconds. Transfer to a plate and set aside.
Wipe out the pan. Add the remaining 1 tbsp oil and heat over medium-high until shimmering. Add the tofu in a single layer and sear undisturbed for 2 to 3 minutes per side, until golden and crisp on at least two sides.
Push the tofu to one side of the pan. Add the garlic, ginger and scallion whites to the cleared space and stir-fry for 20 seconds until fragrant.
Add the soy sauce, oyster sauce, Shaoxing wine and white pepper. Toss everything to coat the tofu evenly, about 30 seconds.
Stir the cornstarch slurry (1 tsp cornstarch plus 2 tbsp water) to recombine and pour it into the pan. Toss until the sauce thickens and clings to the tofu, about 30 seconds.
Fold the scrambled eggs back into the pan along with half the scallion greens. Toss gently just to combine and warm through, 20 to 30 seconds; do not overmix or the eggs will break apart too much.
Remove from heat, drizzle with sesame oil, and top with the remaining scallion greens. Serve over rice if you like.