Preheat the oven to 400F and bring a pot of water to a boil for the pasta. Arrange the broccolini (or broccoli) in one layer on a small baking sheet. Drizzle with oil and sprinkle with salt. Roast the broccoli for 15-20 minutes until soft and browned at the edges. Remove from the oven and set aside.
Meanwhile, cook the pasta to al dente. Right before draining, reserve about a cup of the cooking water.
When the pasta has cooked, add the butter and the minced garlic to the same post used to cook the pasta over medium heat. Let the garlic cook for one minute, then add the pasta, heavy cream, roasted broccoli, scallion, cheese, and half a cup of the pasta water to the pot. Stir vigorously to melt the cheese and form a sauce. Taste the pasta and add salt and pepper to your preference. Stir in a little bit more pasta water if needed to create a sauce that covers the pasta.
Serve the pasta with lemon wedges topped with the sliced chicken. Enjoy!