Bring a large pot of salted water to a boil and cook the pasta until al dente according to package directions; reserve some pasta water then drain.
Slice each chicken breast in half lengthwise to make four thin cutlets, then season both sides with garlic powder, salt, and pepper.
Heat the olive oil and 1 tablespoon butter in a skillet over medium-high heat. When hot, add the chicken and cook 5–6 minutes per side until cooked through and golden; transfer to a plate and let rest, then slice into bite-size pieces.
In the same skillet with any remaining oil/butter, sprinkle in the flour and cook about 1 minute, stirring; if the pan is dry, add 1 tablespoon butter before adding flour.
Gradually pour in the chicken broth and heavy cream while scraping up browned bits, then stir in the pesto, lemon juice, and sun-dried tomatoes. Simmer gently a few minutes until the sauce thickens slightly.
Add the spinach and cook until just wilted, then add the drained pasta and toss to coat, using a splash of reserved pasta water if the sauce is too thick.
Return the sliced chicken to the pan, toss to combine, adjust seasoning with salt and pepper, and serve with freshly grated Parmesan if desired.