Preheat the oven to 350 degrees F. Place the tomatoes in a shallow baking dish. Add the garlic, oil, chili flakes, salt and pepper and toss to combine, arranging the tomatoes so they are cut-side up. Roast the tomatoes 30 minutes, and then remove from the oven.
Cook the linguini in a large pot of lightly salted, boiling water, until firm tender, about 7 minutes. Add the broccoli to the pot and cook 2 minutes more, or until the pasta and broccoli are tender. Drain well, saving 1/2 cup of the cooking liquid. Place the pasta and broccoli in a large serving bowl. Add the reserved cooking liquid, tomatoes and the oil they were cooked in, basil, cheese, salt and pepper; toss to combine and serve.