Bring a large pot of well-salted water to a rolling boil. Add pasta and cook 1 minute less than the package directs (for al dente texture). Drain, do not rinse. Set aside.

Heat olive oil in a 12-inch skillet over medium-high heat. Add chicken in a single layer. Cook undisturbed for 2–3 minutes until golden, flip, and cook another 2–3 minutes until fully cooked (no pink inside).

Push chicken to one side. Add broccoli and let sizzle for 30 seconds. Stir together, add garlic, and cook for 2 minutes until broccoli is bright green and crisp-tender.

Pour in heavy cream and bring to a gentle simmer (look for small bubbles at the edges). Lower heat to medium-low. Slowly add Parmesan, stirring until melted and smooth (about 2 minutes). Add a splash of reserved pasta water if sauce is too thick.

Add cooked pasta to the skillet. Toss to coat evenly. Taste and adjust with salt, pepper, or more Parmesan as needed. The sauce should cling to the noodles without being overly thick.