Mix soy sauce, lime juice, and vinegar. Add tofu and marinate for 20 minutes. Drain and reserve liquid. Heat half of the oil in a non-stick pan over medium heat. In batches, brown tofu on all sides. Set aside. In a mixing bowl, add oyster sauce, chili paste, ketchup, and reserved tofu marinade. Mix cornstarch with enough water to form a thin paste. Whisk into sauce (this is to thicken sauce and dilute). Add remaining oil to a shallow pot over medium-high heat. Add onions and lightly brown. Add mushrooms and cook for 5 minutes. Add bell pepper, ginger, and garlic. Cook for 5 minutes. Add half of sauce and cook for another 5 minutes. Add onions, cooked carrots, and broccolini and cook for 2 minutes. Add more sauce if needed. Serve over rice and place tofu on top.