Bring a large pot of salted water to a boil and cook the pasta according to package directions until just al dente. Reserve 3/4 cup of the cooking water, then drain the pasta and set aside.
While the pasta cooks, pat the chicken dry and season both sides with salt and pepper. Heat 1 tablespoon olive oil in a large skillet over medium-high heat until shimmering.
Add the chicken to the hot pan and sear without moving for 4 minutes, then flip and cook another 4 minutes (internal temperature should reach 165°F). Transfer chicken to a cutting board and let it rest 5 minutes, then slice into bite-size pieces.
Reduce heat to medium and, in the same skillet, add 1 tablespoon olive oil and the minced garlic. Sauté just until fragrant, about 30 seconds, taking care not to brown the garlic.
Pour in the chicken broth to deglaze the pan, scraping up any browned bits. Let it reduce by half for about 2 minutes, then stir in the heavy cream and bring to a gentle simmer.
Lower the heat and whisk in the pesto and grated Parmesan until the sauce is glossy. If the sauce seems too thick, add reserved pasta water a few tablespoons at a time until you reach a silky consistency.
Return the sliced chicken to the skillet and warm through for 1–2 minutes. Add the drained pasta to the pan and use tongs to toss everything together in the sauce so the pasta is thoroughly coated.
Stir in the lemon zest and taste for seasoning; adjust with salt and pepper as needed. Let the whole pan rest off the heat for 2 minutes to let flavors meld.
Serve the pasta in bowls, sprinkle with toasted pine nuts and chopped basil, and finish with an extra shave of Parmesan if desired.