Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs. Drain and transfer to a large bowl.
Meanwhile, heat a large nonstick skillet over medium heat. Add 2 teaspoons vegetable oil, then add half each of the scallions, ginger and garlic; stir-fry until the scallions are wilted, 1 to 2 minutes. Add the shrimp and stir-fry until just cooked through, about 5 minutes. Transfer the shrimp mixture to a plate using a slotted spoon.
Add the remaining 2 teaspoons vegetable oil to the skillet. Add the remaining scallions, ginger and garlic; stir-fry 1 to 2 minutes, then add the broccoli and cook, tossing to coat, 1 to 2 more minutes. Add 1/2 cup broth. Cover and cook until the broccoli is just tender, 5 to 7 minutes.
Whisk the cornstarch and 1 tablespoon water in a bowl, then whisk in the remaining 1 1/2 cups broth, the soy sauce and sesame oil. Return the shrimp to the skillet; add the broth mixture. Bring to a simmer and cook, stirring, until thickened, 1 to 2 minutes. Add the pasta and toss to coat.