Bring a large pot of generously salted water to a boil. Cook the spaghetti according to package directions until al dente. Before draining, reserve about 1 cup of pasta water.
While the pasta cooks, pat the chicken dry and season it with 1/2 teaspoon salt, black pepper, and Italian seasoning.
Heat the olive oil and 1 tablespoon of butter in a large skillet over medium-high heat. Add the chicken in a single layer and cook for 5–7 minutes, stirring occasionally, until golden and cooked through. Transfer the chicken to a clean plate.
Reduce the heat to medium-low. Add the remaining 3 tablespoons of butter to the same skillet, followed by the minced garlic. Cook for about 30 seconds, stirring constantly, until fragrant.
Stir in the lemon zest, lemon juice, and heavy cream. Allow the mixture to gently simmer for 1–2 minutes without bringing it to a vigorous boil.
Reduce the heat to low and gradually stir in the freshly grated Parmesan until melted and smooth. Add a splash of reserved pasta water as needed to create a silky sauce.
Return the cooked chicken to the skillet, then add the drained spaghetti. Toss everything thoroughly until the pasta and chicken are evenly coated in the lemon butter Parmesan sauce.
Taste and adjust with the remaining salt if needed. Add more pasta water, one tablespoon at a time, if the sauce becomes too thick.
Garnish with fresh parsley, additional Parmesan, and optional red pepper flakes. Serve immediately while warm and creamy.