Bring a large pot of salted water to a boil. Add the pasta and cook according to package directions minus 1–2 minutes for al dente. In the last 3 minutes of pasta cooking, add the broccoli florets to the same pot so they steam and cook quickly. Drain, reserving 1/2 cup of pasta water.

While the pasta cooks, heat 1 tbsp of olive oil in a large skillet over medium-high heat. Season diced chicken with salt and pepper. Add to the pan and cook, stirring occasionally, until golden and no longer pink, about 6–8 minutes. Transfer chicken to a plate.

Reduce heat to medium. Add 2 tbsp of butter to the skillet. Once melted, stir in minced garlic and cook for 30–45 seconds until fragrant.

Pour in 1 cup of heavy cream and 1/2 cup of chicken broth, stirring to combine. Simmer gently for 2–3 minutes to thicken slightly.

Stir in 1/2 cup of grated Parmesan until melted and smooth. If the sauce looks too thick, add a splash of the reserved pasta water to loosen it. Taste and season with salt and pepper.

Return the chicken to the skillet, then add the drained pasta and broccoli. Toss everything together until evenly coated in sauce and heated through, 1–2 minutes. If you want a silkier texture, add more reserved pasta water a tablespoon at a time.

Serve immediately, garnished with chopped parsley and extra Parmesan.