Cook pasta according to package directions and transfer to a large serving bowl.
While pasta is boiling, cut chicken breasts into 1/4 to 1/2-inch pieces.
In a large skillet, sauté chicken pieces in the olive oil until cooked through. Add to pasta and mix. Add peppers, garlic and pepper flakes to pan, cook 2 minutes stirring frequently.
Add olives, tomatoes, chicken broth, rosemary and a pinch of salt. Bring the mixture just to a boil. While waiting for liquid to boil, sprinkle cheeses over the pasta and chicken. Pour boiling contents of saute pan over the chicken and pasta and toss.