Bring a large pot of salted water to a boil. Cook the pasta according to package directions until al dente. Reserve 1 cup of pasta cooking water before draining.
While the pasta cooks, season the chicken breasts with salt and pepper. Heat 1 tablespoon of olive oil in a skillet over medium-high heat. Cook the chicken for 5–6 minutes per side, or until fully cooked and internal temperature reaches 165°F (74°C). Remove from skillet and let rest for 5 minutes, then slice into strips.
In a food processor, combine basil, Parmesan cheese, garlic, salt, pepper, and pine nuts (if using). Pulse until finely chopped. With the motor running, slowly drizzle in the remaining olive oil until smooth and emulsified. Set aside.
Return the skillet to low heat. Add the pesto and optional heavy cream (if desired). Stir to combine and warm gently. If the sauce is too thick, add reserved pasta water, 1 tablespoon at a time, until desired consistency is reached.
Add the drained pasta to the skillet with the pesto sauce. Toss well to coat, adding more pasta water if needed. Fold in the sliced chicken.
Serve immediately, garnished with additional Parmesan cheese and a drizzle of olive oil, if desired.
Notes
For best results, use fresh basil and freshly grated Parmesan cheese. Adjust cream amount to your preferred sauce thickness.