Bring a pot of salted water to a boil. Cook the pasta until al dente. Meanwhile, heat the 2 tbsp olive oil in a large skillet over medium-high. Season the chicken with salt and pepper. Sear for about 4 minutes, turning once, until golden and just cooked through. Transfer to a plate.

Reduce the heat to medium and add the 2 tbsp butter to the skillet. Once melted, add the broccoli florets and a pinch of salt, plus 2 tablespoons of water. Cover and steam for about 3 minutes until bright green and crisp-tender. Uncover and cook off any excess water.

Push the broccoli to the edges. Add the minced garlic to the center and cook for 30 seconds, until fragrant. Pour in the 1/2 cup chicken broth and scrape up any browned bits. Stir in the 3/4 cup heavy cream and bring to a gentle simmer. Cook for about 2 minutes until slightly thickened.

Drain the pasta, reserving a little pasta water. Add the pasta, chicken with its juices, and the 1 cup Parmesan to the skillet. Toss over low heat for about 1 minute, until the cheese melts and everything is coated. Add a splash of pasta water if the sauce needs loosening.

Divide among bowls, top with extra Parmesan and black pepper, and serve immediately.