Cook the linguine in salted water until al dente. Save about ⅓ cup of the pasta water, then drain and set the pasta aside.
Season the chicken with Italian seasoning, garlic powder, onion powder, ½ teaspoon salt, and black pepper.
Heat the olive oil and 2 tablespoons butter in a large skillet over medium-high heat. Add the chicken and cook for 6 to 8 minutes, turning a few times, until browned on the outside and cooked through.
Add half of the garlic to the skillet and stir for about 30 seconds so the chicken gets coated in the garlic butter. Transfer the chicken to a plate and keep warm.
Lower the heat to medium. Add the remaining 2 tablespoons butter to the same skillet, then stir in the rest of the garlic for about 30 seconds.
Pour in the heavy cream and chicken broth. Add the remaining ¼ teaspoon salt and stir until warmed through.
Mix in the Parmesan and mozzarella a little at a time, stirring until the sauce is smooth, thick, and creamy.
Add the linguine and toss until the sauce coats the pasta well. If needed, add a splash of reserved pasta water to loosen it slightly.
Spoon the creamy linguine onto plates and arrange the garlic butter chicken beside it. Finish with chopped parsley and serve warm.