Bring a large pot of well-salted water to a boil. Cook the pasta according to the package directions until al dente, adding the broccoli during the final 2–3 minutes. Reserve at least 1/2 cup pasta water, then drain the pasta and broccoli.

Season the chicken pieces with about 1/4 teaspoon kosher salt and the black pepper. Heat the olive oil in a large skillet over medium-high heat, then cook the chicken for 5–7 minutes, stirring occasionally, until browned and cooked through to 165°F.

Reduce the skillet heat to medium and add the minced garlic. Cook for about 30 seconds, just until fragrant, taking care not to let it brown.

Pour in the cream or half-and-half and bring it just to a gentle simmer. Do not let the mixture boil vigorously, since a moderate temperature helps keep the sauce smooth.

Stir in the pesto and Parmesan, then add the drained pasta and broccoli. Toss everything together until the sauce coats the chicken, vegetables, and pasta.

Add reserved pasta water 1 tablespoon at a time until the sauce is glossy and creamy. Taste and adjust the salt, then add lemon juice or red pepper flakes if desired and serve with extra Parmesan.