How to Make It
Peel the broccoli stems, removing all the tough, dark green skin and any stringy parts. Separate and wash the florets and stems.
Bring a large pot of water to a boil and add 2 tablespoons of coarse salt. Add the thick broccoli stems and cook for 7 minutes, then add the florets. When the water returns to a boil, cook for 12 minutes longer. Drain well.
Wash and drain the scallops. Trim off the small beige filament from the side of each one. Slice the scallops crosswise into thin rounds; pat dry.
Separate the larger broccoli florets into smaller ones and slice all of the stems into thin rounds.
Put 6 tablespoons of the butter and the onion in a large skillet. Turn the heat to moderately high and cook, stirring often, until the onion is softened but not browned, 3 to 5 minutes. Add all the broccoli and stir gently to coat. Reduce the heat to moderate and cook for 6 minutes, stirring occasionally; set aside.
Fill a large pot with 4 quarts of water and bring to a boil. Add 1 1/2 tablespoons of coarse salt, cover and return to a boil. Add the pasta and stir rapidly with a wooden spoon. Cover and bring back to a boil, then uncover and cook the pasta, stirring frequently, until it is al dente.
Meanwhile, add the sliced scallops, thyme and chile to the broccoli and season with salt. Cook briefly until the scallops turn white, 2 to 3 minutes. remove from the heat and stir in the cheese.
When the pasta is nearly done, add the remaining 2 tablespoons of butter of the scallops and broccoli. return the pan to moderate heat and stir gently. Drain the pasta and add it to the skillet. Toss thoroughly for no more than 20 seconds, then transfer the pasta to a warm bowl and serve at once.