Cook the rice according to package directions, and once cooked fluff with a fork. Spread out a rimmed baking sheet to cool.
While the rice is cooking heat 1 tablespoon of the olive oil in a medium skillet over medium heat. Add the Vidalia onion, season with salt and pepper and cook for about 20 minutes, stirring often, until the onions are tender and caramelized. Adjust the heat so that the onions are cooking and browning slowly – that’s how they will get even sweeter and gently caramelized. Stir in the parsley.
Return the skillet to the heat. Add another tablespoon oil, then add the broccolini and red pepper flakes and sauté for 5 minutes or so until the broccolini is crisp tender. Spread out on a plate to cool to room temperature (this will help it from steaming and overcooking).
In a large bowl stir the cooled rice with the onion-parsley mixture, the sautéed broccolini, and the slivered red onion. Taste and adjust seasoning as needed. Add more olive oil by the tablespoon if the mixture seems dry.