Bring a large pot of water to boil for the pasta. Once it reaches a full boil, add the pasta and continue to boil just until the pasta is tender, but still slightly firm in the center (7-8 minutes). Once the pasta is tender, add the frozen broccoli florets to the water with the pasta and continue to cook for 2-3 more minutes, or until the broccoli is tender. Drain the pasta and broccoli in a colander.
While the pasta is cooking, cut the chicken breast into ½-inch pieces. Heat one tablespoon of olive oil in a large skillet over medium heat. Once hot, add the chicken and continue to cook until the outside of the chicken pieces are white, but they're still tender in the center (they will continue to cook as more ingredients are added).
Add the chicken broth to the skillet with the chicken and allow it to heat through. Once hot, add the cream cheese (cut into small pieces) and pesto. Continue to stir and cook until the cream cheese has melted and created a smooth, creamy sauce with the pesto and chicken broth.
Finally, add the cooked and drained pasta and broccoli to the skillet and stir to coat it in the cream sauce. If the sauce becomes too dry or thick, add another splash of chicken broth to loosen it up. Serve hot.