Press tofu between paper towels with a weight for 15 minutes. Cut into cubes and pan-fry in vegetable oil over medium-high heat until golden and crispy on all sides, about 8 minutes. Set aside.
Heat sesame oil in the same pan. Add minced garlic and ginger, cook for 30 seconds until fragrant.
Add broccolini and stir-fry for 3–4 minutes until tender-crisp and lightly charred.
Whisk together oyster sauce, soy sauce, and rice vinegar. Pour into the pan with the broccolini and toss to coat.
Return tofu to the pan and toss gently to combine. Scatter sesame seeds and sliced spring onions on top.