Pat the chicken dry. In a bowl, toss it with Italian seasoning, black pepper, and 1 teaspoon of the kosher salt (reserve the remaining salt for the pasta water).
Set a large pot of water over high heat for the pasta. While it heats, place a wide skillet over medium-high heat and add the olive oil and 1 tablespoon of butter.
Sear the chicken in the hot skillet in an even layer, undisturbed for 2 minutes, then cook, stirring occasionally, until golden and just cooked through, 5–7 minutes total. Transfer to a plate and tent loosely with foil.
Reduce heat to medium and add the remaining 1 tablespoon butter. Stir in garlic and crushed red pepper flakes; cook 45–60 seconds until fragrant, then pour in the chicken broth, scraping up any browned bits. Simmer 2 minutes to slightly reduce.
Lower the heat to medium-low and stir in the heavy cream. Maintain a gentle simmer, stirring occasionally, until lightly thickened, 4–5 minutes. Sprinkle in the Parmesan gradually, stirring until smooth and glossy. Taste and adjust salt if needed.
Salt the boiling water with the remaining kosher salt and cook the penne until just shy of al dente per package time. Reserve 3/4 cup pasta water, then drain.
Remove the skillet from the heat and stir in the basil pesto and lemon zest. Add a splash of reserved pasta water (start with 1/4 cup) to loosen the sauce if needed and return the skillet to low heat.
Add the drained pasta to the skillet and toss over low heat for 1–2 minutes so it finishes cooking and absorbs the sauce. Fold in the chicken and any plate juices; warm through for 1 minute. Adjust consistency with additional pasta water to taste.
Serve immediately with extra Parmesan if you like.