Cook linguine following package instructions, about 9 minutes. Add broccoli during last 5 minutes of cooking; drain.
In a small bowl, whisk together the broth, soy sauce, ketchup, sugar and hot sauce.
Place the broth mixture in a large skillet and bring to a simmer. Add scallops and water chestnuts and simmer for 4 minutes, turning halfway through, or until cooked through. Remove from heat and stir in sesame oil.
Place cooked pasta and broccoli in the skillet and toss with the scallops. Serve immediately.