Preheat oven to 400ºF. Toss onions with vegetable oil, and salt and pepper to taste. Place on a rimmed baking sheet and roast until softened and starting to brown (about 15 minutes). Reserve.
Bring a large pot of salted water to a boil. Cook the broccoli until crisp tender (about 3 minutes). Remove broccoli from water, saving the water to cook the pasta. Reserve broccoli.
Cook rotini according to package directions. Drain and reserve.
Heat a sauté pan over medium heat. Add the olive oil and broccoli. Cook to brown the broccoli (about 2 minutes). Add the garlic and cook until aromatic (about 1 minute). Stir in the chicken, pasta, and onion. Deglaze the pan with the white wine. Cook for a couple of minutes to cook off the alcohol and heat chicken through. Taste and adjust seasoning to taste.
Divide between 4 bowls. Sprinkle with the Parmigiano.