Bring a large pot of water to a rolling boil and cook the lo mein noodles 3–4 minutes or according to package directions; drain and set aside.
In a small bowl whisk together the lime juice, brown sugar, soy sauce, Sriracha, grated ginger, and 1 teaspoon sesame oil; set the sauce aside.
Heat the non-stick skillet over medium and melt the butter. Add the crushed red pepper flakes, pour in the beaten eggs, and gently scramble until just set, about 2–3 minutes; transfer the eggs to a plate and keep warm.
Add 1 tablespoon sesame oil to the pan and raise heat to medium-high. Add the shrimp and cook 1–2 minutes per side until opaque and pink; remove with a slotted spoon and set aside.
Increase heat to high, add the frozen broccoli and red pepper strips plus the remaining sesame oil, and stir-fry until just charred and heated through, about 3–4 minutes. Stir in the minced garlic and cook for 1 minute until fragrant.
Add the drained noodles to the skillet along with the cooked eggs, shrimp, and the prepared sauce. Reduce heat to medium-low and toss well to coat and heat through, about 1 minute; taste and add salt only if needed.
Serve immediately.