Preheat the oven to 425° F. Peel the onions and remove their tops and tails. Slice each one crosswise into 3 slices, about 3/4 inch thick, and place them on a parchment-lined baking sheet. (You can skip the parchment, but you'll spend more time cleaning.) Drizzle the slices with olive oil and smoosh the oil around with your (clean) hands to coat evenly. (You can also use a pastry brush; I don't own one.) Sprinkle with a big pinch of salt and a few grinds of pepper. Roast for 40 minutes, until the onions begin to brown and caramelize, and are soft but not totally slouchy. (If you want more color out of them, stick them under the broiler for a minute or two.) Set them aside to cool just a bit.
While the onions cook, combine all of the salsa ingredients in a small bowl, add 1/4 teaspoon salt, stir, and set aside.
To serve, put the arugula and parsley in a large bowl. Toss with about half the salsa, then nestle in the onion slices, dollop on the cheese, and top with the rest of the salsa. Serve.