Position racks in the upper and lower thirds of the oven and heat the oven to 425°F.
Spread the chicken and cauliflower in single layers on separate large rimmed baking sheets. Toss each with 1 Tbs. of the oil, 1 tsp. salt, and several grinds of pepper. Roast with the chicken in the top third of the oven, switching positions and rotating the pans halfway through, until the cauliflower is golden and crisp-tender, about 20 minutes, and the chicken reaches 165°F, 20 to 25 minutes. Tent the chicken with foil to keep warm.
Meanwhile, combine the onion with the vinegar and 1/4 tsp. salt in a small bowl. Let sit until the onion has mellowed and turned a vibrant pink, 10 to 12 minutes.
Drain the onion over a small bowl. Whisk the mustard, 1/4 tsp. salt, and a few grinds of pepper into the vinegar. Slowly whisk in the remaining 3 Tbs. oil.
Slice the chicken crosswise 1/2 inch thick. In a large bowl, combine the chicken, cauliflower, onion, chickpeas, olives, and parsley. Toss with the dressing, and then season to taste with salt and pepper. Add the arugula and gently toss again before serving.