Bring a large pot of water to a boil for pasta. When it comes to a full boil, salt the water and add broccolini or broccoli and par-boil about 3 minutes. Remove with a slotted kitchen utensil and cool in an ice water bath, then drain. Add pasta and cook to al dente.
Heads up: you will need a couple of ladles starchy cooking water just before draining pasta.
Place cheese in the bottom of a shallow serving bowl.
Meanwhile, heat EVOO, 3 turns of the pan, over medium heat and add chili pepper and garlic, sauté a couple of minutes. Then add thyme and stir, then add broccolini or broccoli and heat through. Add pasta and cheese and toss with a little starchy water to combine and coat pasta.
Garnish with additional grated cheese.