Place noodles in a large heatproof bowl and cover with hot tap water. Soak until loose and pliable, about 30 minutes. Separate noodles by hand and drain; set aside. Meanwhile, place tofu cubes on a paper-towel-lined plate; set aside. Combine soy sauce, 1/4 cup of the water, and the brown sugar in a small bowl; set aside. Heat remaining 1/4 cup water in a large nonstick frying pan with a tightfitting lid over medium-high heat until simmering. Add broccolini, cover, and cook until just beginning to soften, about 2 minutes. Remove the lid and cook, stirring often, until water has evaporated and broccolini is just tender when pierced with a knife, about 2 minutes more. Remove broccolini to a plate and set aside. Return the pan to medium-high heat, add oil, and heat until shimmering. Add reserved tofu and sauté, turning occasionally, until golden brown on all sides (be careful—the oil will splatter), about 8 to 10 minutes. Reduce heat to medium, add garlic, and cook until just beginning to color, about 15 seconds. Add reserved soy sauce mixture, noodles, and broccolini and cook, tossing gently, until noodles are soft, warmed through, and coated with sauce, about 4 minutes. Push noodle mixture to one side of the pan, add eggs, and scramble until eggs begin to set, about 30 seconds. Let eggs cook undisturbed until solid, about 1 minute more. Remove from heat and toss to evenly combine all ingredients. Serve immediately.