Begin by bringing a large pot of heavily salted water to a boil.
While your water comes to a boil, heat a large cast iron skillet with olive oil, over high heat. Once pan is hot, add the chicken in a single layer. Season with salt and pepper and cook, undisturbed, until the chicken becomes white around the edges. Flip the chicken and cook on the second side until cooked through, about 4 minutes.
Once the water is boiling, add the pasta. When there are 90 seconds remaining on the pasta, add the broccoli.
Remove cooked chicken from the pan and set aside.
With the heat still on high, add the wine and deglaze the pan. Allow the wine to bubble up and slightly reduce, while you scrape the bottom of the pan, releasing any browned bits that formed from cooking the chicken.
When the wine has reduced slightly, add the butter and cream. Cook for two minutes over medium high heat, until the sauce begins to thicken.
Add ¼ cup of the cheese and stir to combine. Once the cheese has melted, taste the sauce and add salt and pepper if needed. Continue cooking the sauce on the lowest heat while the pasta finishes cooking.
Once pasta and broccoli have finished cooking, add them to the sauce. Scoop the pasta and broccoli, using a strainer or slotted spoon, directly from the water into the sauce. Some of the pasta water will transfer into the sauce, helping to thicken and emulsify the sauce. Top with remaining ¼ cup cheese and parsley. Stir to combine and serve with additional Parmesan cheese.