Put ribs and 5 cups water in a wide 8- to 10-qt. pot and bring to a rolling boil over medium-high heat (ribs will not be completely covered). Skim any foam that rises to the top and boil for 5 minutes. Reduce heat to medium-low and simmer 1 hour, partially covered and stirring occasionally, skimming any foam and fat that rise to the top.
Coarsely shred apple and add to pot with soy sauce, sherry, sugar, pepper, ginger, garlic, and shallot. Continue to cook, partially covered and stirring occasionally, until ribs are tender when pierced with tip of a small knife, about 1 hour; skim fat during cooking.
Meanwhile, cut carrots into 2-in.-long pieces. Cut purple-top turnips in half, then into 1-in.-thick wedges (leave Tokyo turnips whole). Cut 4 green onions into 2-in.-long pieces and thinly sliver remaining onions; set slivered onions aside.
Add carrots, turnips, and the 2-in. pieces of green onion to pot. Continue to cook, partially covered and stirring occasionally, until vegetables are tender, 20 to 30 minutes. Carefully tip pot and skim off as much fat as possible from liquid with a wide metal spoon.
Stir in sesame seeds and sesame oil. Serve sprinkled with slivered green onions.
Make ahead: Through step 2 up to 1 day, chilled; lift off and discard fat before reheating.
Tokyo Turnips: These sweet, tender little globes, available at farmers' markets, are starting to show up in grocery stores.