Bring a large pot of salted water to a boil. Cook spaghetti according to package directions, adding broccoli to the pasta water during the last 20 seconds of cooking. Meanwhile, heat olive oil in a large skillet over high heat. Season chicken on all sides with salt and pepper and cook, browning well, for about 10 min. Drain pasta and broccoli, reserving 1/2 cup (125 mL) of the cooking water.
Add the onion and garlic to the chicken and cook a few minutes until fragrant. Add tomatoes, oregano, olives and reserved pasta water. Simmer for a few minutes to reduce and thicken slightly. Mix in pesto.
Divide the pasta evenly among 4 plates or bowls and top with the chicken mixture. Sprinkle with Parmesan cheese and serve.