Bring a large pot of water to a boil and add 1/2 teaspoon salt. Add the penne and cook according to package directions until al dente.
4 minutes before the pasta is done, add the broccoli florets to the boiling water and cook until bright green and tender; remove the broccoli with a slotted spoon and set aside.
Reserve 1/2 cup of the pasta cooking water, then drain the pasta in a colander.
While the pasta cooks, heat the olive oil and butter in a large deep skillet over medium heat until the butter melts.
Add the chicken pieces to the skillet and season with 1/2 teaspoon salt, 1/4 teaspoon black pepper, 1/2 teaspoon garlic powder, 1/2 teaspoon paprika, and 1/2 teaspoon Italian seasoning. Cook, stirring often, until the chicken is browned and cooked through; remove the chicken and set aside.
Add the minced garlic to the skillet and sauté about 1–3 minutes until fragrant. Stir in the cream cheese and cook until it melts and combines with the garlic.
Whisk the cornstarch into the whole milk until smooth, then slowly pour into the skillet, stirring constantly. Cook until the sauce thickens, adjusting thickness with reserved pasta water a little at a time if needed.
Return the cooked chicken and broccoli to the skillet, add the drained pasta, and gently toss to coat everything in the sauce.
Stir in 1/2 cup Parmesan cheese and taste for seasoning. Serve immediately with extra Parmesan if desired.