Trim fat from pork, leaving 1/8-inch thick layer. Sprinkle pork with salt and pepper. In large Dutch oven, heat oil over medium-high heat; brown pork all over. Transfer to plate.
Reduce heat to medium and drain off fat; cook onion, stirring often, until golden, about 4 minutes. Add milk, stirring to scrape up any brown bits.
Return pork and any accumulated juices to pan; bring to simmer. Cover and braise in 300°F oven, basting every 30 minutes and turning once with 2 wooden spoons, until pork is very tender, 2 to 2 ½ hours.
Transfer to cutting board and tent with foil; let stand for 10 minutes. Cut strings from pork and slice across the grain.
Meanwhile, skim fat from sauce and bring to boil over high heat; boil until reduced to 2 ½ cups, about 10 minutes. Transfer to blender or food processor and purée until smooth. Blend in lemon juice. Stir in parsley. Serve sauce with sliced pork. Add cipollini (if using) to pan during last hour of braising. With slotted spoon, transfer to platter with pork before puréeing sauce.