Cook pasta according to package directions. Rinse and drain well in a colander.
In a medium bowl, soak sun-dried tomatoes in hot water to cover. Let stand until softened, about 5 minutes. Drain tomatoes; transfer to a cutting board and slice crosswise into strips.
In a skillet, heat 1 T. oil over medium heat. Add chicken, garlic and rosemary; cook, stirring occasionally until chicken is no longer pink on the outside, about 5 minutes.
Add broccoli florets, green onions, sun-dried tomatoes, chicken broth, 1 teaspoon salt and remaining oil. Cover, cook until broccoli is bright green and crisp-tender, about 5 minutes.
Transfer cooked pasta to a large serving bowl. Add chicken and broccoli mixture, 1/2 cup Parmesan, pepper and remaining salt; toss to coat.
Serve with remaining Parmesan on the side.