Bring a large pot of salted water to a boil and cook the pasta until al dente according to package directions; reserve about 1/2 cup pasta cooking water if desired, then drain.
While the pasta cooks, heat the olive oil in a skillet over medium-high heat.
Add the chicken strips to the skillet and sauté for 3–4 minutes, stirring, until the outside is no longer pink (chicken will finish cooking in the sauce).
Stir the pesto and heavy cream into the skillet with the chicken, reduce heat to medium, and simmer for about 5 minutes until warmed and slightly thickened.
Add the drained pasta to the skillet and toss to coat with the sauce; add a splash of reserved pasta water if needed to loosen the sauce.
Season with salt and pepper to taste and serve topped with freshly grated Parmesan if desired.