Press the tofu: Drain the firm tofu and press it between paper towels for 15 minutes to remove excess water.
Prepare the sauce: In a small jug, whisk together soy sauce, ginger paste, brown sugar, and rice vinegar.
Crisp the tofu: Cut the pressed tofu into 1-inch cubes. Toss with cornstarch and salt until evenly coated. Heat half the vegetable oil in a non-stick pan over medium-high heat and fry tofu until golden brown and crispy on all sides (5-7 minutes). Remove and set aside.
Stir-fry veggies: Add remaining oil to the pan. Stir-fry mushrooms until they soften (about 2-3 minutes). Add broccolini and red bell pepper, sprinkle with salt, and cook for another 3-5 minutes until tender-crisp.
Add aromatics and sauce: Stir in minced garlic and crushed red pepper flakes; cook for 1 minute until fragrant. Pour in the prepared sauce and bring to a simmer.
Combine and serve: Add the crispy tofu back to the pan with the stir-fried veggies and sauce. Toss gently to coat everything. Serve immediately over cooked rice.