As the quail eggs cook in the sauce, prepare the broccolini. Heat the oil in a skillet over medium-high. Once hot, add the onion and cook for 7 minutes, stirring regularly, until softened and beginning to brown. Add the broccolini and cook, stirring often, for about 10-12 minutes or until the broccolini is tender-crisp and bright green. Season with salt and pepper.