Bring a large pot of water to a boil. Add the salt to the water, return the water to a boil and cook the pasta according to package directions, stopping two minutes or so before it is fully cooked. Drain the pasta.
Meanwhile, in a small bowl or container, combine the broth, soy sauce, cornstarch, sugar, sesame oil, and pepper.
In a large skillet or wok over high heat, heat 1 ½ teaspoons of the oil, then add the shrimp and sauté until the shrimp are almost (but not quite!) cooked, about 2 minutes. Transfer them to a plate and set aside. Add the remaining 1 ½ teaspoons of oil to the skillet, add the garlic, broccoli, carrots, and red pepper and sauté for 3 to 4 minutes until everything starts to soften.
Return the shrimp to the pan along with the drained pasta and the sauce. Toss (tongs are great for this) over the high heat until the noodles absorb some of the sauce, everything is well mixed, and the rest of the sauce is slightly thickened, another 2 minutes or so. Serve hot.
Notes
This is a nice opportunity to let kids pick the vegetables they like to include in this recipe. They can also peel the carrots, measure and mix up the ingredients for the sauce, and peel the shrimp (wash hands thoroughly before and after!)
You can use either long spaghetti-thickness Chinese noodles, which might be labeled lo mein noodles, or you can use spaghetti. Just cook them according to package directions and make sure not to cook them past just tender.
You can also bump up the ratio of vegetables and shrimp to noodles, resulting in a meal that looks and tastes great and does the right thing by your family nutritionally. My kids never flinch at a vegetable when it's in the context of a stir fry.