Cook the spaghetti according to the label directions and drain, reserving 1/2 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 large skillet over medium heat. Add the garlic, broccoli, and half the salt and cook until the garlic is lightly browned and the broccoli is fork-tender, 4 to 5 minutes. Add the pasta, the shredded chicken, the reserved pasta water, the remaining 1/4 teaspoon salt, and a few grinds of pepper. Cook until heated through, about 3 minutes. Add the red pepper flakes and toss well. Drizzle with the remaining oil before serving. Serve with Crispy Garlic Bread.