Cook pasta in a large pot of salted boiling water for 1 minute less than al dente, according to package instructions (that was about 9 minutes total for my box of penne). During the final 3 minutes of cooking time, add the broccoli florets to the boiling water with the pasta. Drain.
Meanwhile, heat olive oil in a large skillet or Dutch oven over medium-high heat. Season the chicken with salt and pepper on both sides. Cook the chicken in the oil until browned on both sides and juices run clear, about 5-7 minutes per side. Remove the chicken to a cutting board, let rest for 5 minutes, and then slice into thin strips.
In the same pan, melt the butter over medium heat. Add the garlic and stir until fragrant, about 30 seconds. Stir in asparagus, increase the heat to medium-high, and cook for 1 minute. Add the zucchini and bell pepper, cook for 1 minute. Stir in the tomatoes and cook for 1 more minute.
Add the chicken broth, increase the heat and bring the mixture to a boil.
Reduce heat to low so that the liquid is at a gentle simmer. Stir in the cream and Parmesan until smooth.
Add the cooked pasta, broccoli, peas, chives and basil to the sauce. Let it bubble and thicken slightly, stirring frequently, until all of the ingredients are warmed through, about 2 more minutes. Taste and season with salt and pepper, if necessary. Stir in the chicken and serve immediately.