Cook pasta according to package directions. Add broccoli florets during the last 4 minutes of cook time. Drain well.
While pasta is cooking, season chicken with salt and pepper.
Melt 1 tablespoon of butter in a large skillet over medium high heat (I used a 12 inch skillet). Add chicken and cook for approximately 6 to 7 minutes (3 to 3 ½ minutes on each side), or until chicken is no longer pink and fully cooked. Transfer chicken to a plate and set aside.
In the same skillet over medium heat, melt the remaining ⅓ cup of butter. Add garlic and sauté (stirring constantly), until fragrant (about 30 seconds). Add cream cheese cubes and Italian seasoning. Whisk to smooth out any large lumps.
Slowly pour in milk, about ¼ cup at a time, while continuing to whisk.
Add Parmesan cheese and whisk or stir well to combine. Continue to cook and stir for 2 to 3 minutes, or until sauce reaches desired thickness.
Remove from heat and add pasta, broccoli and cooked chicken to skillet with sauce. Toss to coat evenly. Season with additional salt and pepper, if desired.
Serve sprinkled with grated Parmesan cheese and fresh chopped parsley, if desired.
If reheating and sauce is too thick, add milk to reach desired consistency.