Slice each chicken breast in half widthwise to make 4 thinner pieces and lightly pound to even thickness; season both sides with salt, pepper, and half of the Italian seasoning.
Bring 12 cups of water to a boil in a large pot, add 1 tablespoon salt, and cook the pasta 1 minute less than package instructions; reserve 1 cup pasta cooking water, then drain the pasta.
Heat 2 tablespoons butter in a large heavy-bottomed skillet over high heat; add the chicken in a single layer and sear 2–3 minutes per side until cooked through (160°F) and juices run clear, drizzling about 1 tablespoon oil from the sun-dried tomato jar over the chicken as you flip; transfer chicken to a plate and tent with foil.
Reduce heat to medium and add the remaining 1 tablespoon butter; when melted, add the minced garlic and cook 30 seconds until fragrant.
Pour in 1/2 cup chicken broth and scrape up browned bits; when mostly evaporated, add the remaining 1/2 cup broth and stir.
Stir in the heavy cream, grated Parmesan, chopped sun-dried tomatoes, and the remaining Italian seasoning; simmer gently for 3–5 minutes until slightly thickened.
While the sauce thickens, dice or shred the cooked chicken.
Add the spinach, cooked chicken (with any juices), and drained pasta to the sauce; toss with tongs 1–2 minutes until spinach wilts and the sauce coats the pasta, adding a splash of reserved pasta water if the sauce is too thick.
Stir in the pesto if using, toss again, then taste and adjust seasoning as needed.
Serve immediately topped with extra grated Parmesan and the chopped parsley or basil.