Heat oven to 375°F. Bring a large pot of well-salted water to a boil.
Pat chicken dry and, if thick, slice breasts in half horizontally to create even cutlets. Season all over with Italian seasoning, kosher salt, and black pepper.
Sear the chicken: Warm a large oven-safe skillet over medium-high heat with the olive oil. Cook chicken until deeply golden and just cooked through, about 4–5 minutes per side (165°F internal). Transfer to a board to rest; then cut into bite-size pieces, saving any juices.
While the chicken cooks, boil the penne 2 minutes shy of al dente. Reserve 1/2 cup pasta water, then drain.
Build the sauce: Return the skillet to medium heat and add butter and the sun-dried tomato oil. Stir in garlic and chopped sun-dried tomatoes; cook until fragrant, about 60 seconds. Sprinkle in red pepper flakes.
Pour in heavy cream and chicken broth, scraping up any browned bits. Simmer gently, stirring, until slightly thickened, 3–4 minutes. Gradually whisk in 3/4 cup Parmesan until smooth. Stir in lemon zest and fold in spinach to wilt, 1–2 minutes. Taste and adjust seasoning.
Combine and adjust: Add the drained pasta, chopped chicken, and any resting juices to the skillet. Toss until coated, loosening with reserved pasta water a splash at a time until the sauce clings and looks glossy.
Bake: Transfer the mixture to a 9×13-inch baking dish (or keep in oven-safe skillet). Sprinkle with the remaining 1/4 cup Parmesan and bake until bubbling at the edges, 8–10 minutes. For browned spots, broil 1–2 minutes, watching closely.
Rest 5 minutes so the sauce settles, then serve hot.