Prep the chicken: Pat the chicken dry and season with salt, pepper, and half the Italian seasoning. This helps build flavor from the start.
Sear the chicken: Heat olive oil in a large skillet over medium-high heat.
Add the chicken in a single layer and cook 5–7 minutes, stirring once or twice, until browned and cooked through. Transfer to a plate and keep warm.
Sauté the veggies: Reduce heat to medium. Add butter to the skillet.
When melted, add broccoli and a pinch of salt. Cook 3–4 minutes, stirring occasionally. If using zucchini, add it and cook 2 minutes more until crisp-tender.
Remove veggies to the plate with the chicken.
Build the sauce: In the same skillet, add garlic and cook 30 seconds until fragrant. Pour in chicken broth and scrape up any browned bits. Stir in heavy cream and cream cheese.
Whisk until the cream cheese melts and the sauce is smooth.
Thicken and season: Lower heat to medium-low. Stir in Parmesan, remaining Italian seasoning, and red pepper flakes if using. Simmer 3–4 minutes, stirring, until slightly thickened.
Taste and adjust salt and pepper.
Combine: Return chicken and veggies (and any juices) to the skillet. Toss to coat in the sauce and warm through, about 2 minutes. If the sauce is too thick, splash in a bit more broth.
Finish and serve: Sprinkle with parsley and extra Parmesan.
Squeeze a little lemon over the top if you like a bright finish. Serve hot.