Bring a large pot of salted water to a boil. Drop in pasta and cook for 8 to 10 minutes or until al dente. Drain.
Meanwhile, put olive oil and minced garlic in large skillet. Cut chicken into small pieces and brown in skillet. Add tomatoes and rosemary and stir together. Parboil broccoli (by partially cooking it in boiling water, just to soften) and add it to the chicken mixture. Pour cream over all, stir together and bring to a simmer. Add chicken mixture to cooked pasta and toss to coat. Top with cheese and serve.
