Cook the spaghetti according to the package directions. Drain, reserving ½ cup of the cooking water.
Meanwhile, cut off the broccoli stalk, peel it, and thinly slice it crosswise. Cut the remaining broccoli into small florets.
Heat half the oil in a skillet over medium heat.
Add the broccoli, garlic, and ¼ teaspoon salt and cook until garlic is lightly browned and broccoli is fork-tender, 4 to 5 minutes.
Add the pasta, reserved pasta water, chicken, remaining ¼ teaspoon salt, and a few grinds of pepper. Cook until heated through, about 3 minutes.
Add the crushed red pepper and toss well. Drizzle with the remaining oil before serving.