Boil the pasta: Bring a large pot of salted water to a rolling boil.
Cook the pasta until just shy of al dente. Reserve 1 cup of pasta water before draining.
Season the chicken: Pat the chicken dry. Season with 1 teaspoon salt, 1/2 teaspoon black pepper, and a pinch of red pepper flakes if you like.
Sear the chicken: Heat olive oil and butter in a large skillet over medium-high. Add the chicken in a single layer and cook until golden and cooked through, 5–7 minutes.
Transfer to a plate.
Sauté the garlic: Lower the heat to medium. In the same skillet, add a touch more oil if needed. Sauté garlic for 30–45 seconds until fragrant, not browned.
Build the sauce: Pour in the cream and 1/3 cup broth or reserved pasta water.
Simmer gently for 2–3 minutes to thicken slightly.
Add pesto and cheese: Stir in the pesto and Parmesan until smooth. If it’s too thick, add more pasta water a splash at a time. Aim for a glossy, pourable sauce that coats the back of a spoon.
Brighten it up: Add lemon zest and 1 teaspoon lemon juice.
Taste and adjust with more salt, pepper, or lemon as needed.
Add the chicken and greens: Return chicken (and juices) to the pan. If using spinach or peas, stir them in now. Cook 1–2 minutes until greens wilt and chicken is heated through.
Toss with pasta: Add drained pasta to the skillet.
Toss until every piece is well coated, adding more pasta water if needed to keep the sauce silky.
Finish and serve: Remove from heat. Top with extra Parmesan, a few basil leaves, and a final crack of black pepper. Serve warm.