Put the fresh pork ribs in a pot of cold water with ginger slices and cooking wine, bring to a boil over high heat, skim off the foam, and boil 2 more minutes before scooping them out.
Wash the blanched ribs several times in warm water until every trace of surface foam is gone.
Add boiling water to a clay pot, add the ribs and ginger slices, cover, bring to a boil over high heat, then turn to low and simmer about 40 minutes — use the braising time to prep the vegetables.
Peel and cut the potatoes into thick batons, cut the garlic sprouts into segments, slice the onion, cut the dried chilies into segments, and mince the garlic.
Heat oil in a wok, add the potatoes, and pan-fry them — you can cover the wok with a lid during frying so they cook through more easily.
Fry until the surface is golden brown, then scoop them out and set aside.
After 40 minutes the ribs are cooked through — they will still be seared, so they don't need to be braised fall-apart tender, just chewy-tender.
Scoop the braised ribs out of the pot.
Heat a little oil in a wok, add the braised ribs, and stir-fry over medium-low heat until the surface is faintly golden, then scoop them out.
Without washing the wok, add a little oil, add the minced garlic and dried chilies and fry until fragrant, then add two spoonfuls of dry pot sauce and stir over low heat until aromatic — if you want heat, add fresh bird's eye chilies; these dried chilies are mild enough for kids.
Add the onion and garlic and stir-fry for a few seconds.
Add the ribs and potatoes, stir-fry over high heat until the onion is just cooked.
Add a splash of light soy sauce, keep tossing over high heat to coat evenly, then turn off the heat right away.
Sprinkle on toasted white sesame seeds and toss — add them if you have them, skip them if you don't.
Every rib is coated in the mildly spicy, savory dry pot sauce, crisp outside and tender inside — no salt needed since the dry pot sauce and soy sauce are already seasoned.
Don't waste the braising liquid — bring it to a boil over high heat, add two spoonfuls of salt, and it's ready to drink as is.
Add amaranth greens and simmer about 1 minute before serving — greens blanched in rib broth make both the vegetable and the soup sweet and fresh.