1. Heat 1 tablespoon oil in a large skillet over medium-high. Add eggs and 1/2 teaspoon salt; cook, undisturbed, until edges are set, about 30 seconds. Lift edges with a spatula; tilt pan so runny parts slide to bottom. Cook until golden in places on bottom but still wet on top, about 1 minute. Slide onto a cutting board; let cool slightly, then roll up like a cigar and slice into 1/2-inch-thick "noodles."

2. Wipe skillet clean; add remaining 2 tablespoons oil and place over high heat. When oil shimmers, add ginger, garlic, Broccolini, mushrooms, and half of scallions. Cook, stirring, until vegetables are golden in places, about 5 minutes. Stir in chile and 1/4 teaspoon pepper; cook 30 seconds. Add soy sauce, lime juice, egg "noodles," and remaining scallions; toss to combine. Serve.