Preheat your oven to 200°C (390°F).
Pop a tablespoon of neutral cooking oil into an oven pan, spread your chicken pieces out evenly, and roll them in the oil to ensure they’re fully covered. Squeeze the juice of the limes onto both sides of the chicken pieces, and pop the squeezed lime quarters beneath each chicken piece. Season the chicken with a pinch of sea salt and cracked black pepper and roast in the oven for 15-20 minutes until light brown.
Fry the roughly-chopped shallots or white onion in a frying pan until just translucent, then add the sliced garlic cloves until light brown, taking care not to burn either.
Add the remaining tablespoons of cooking oil to a big soup pot, along with the fried shallots/onion and garlic, chopped carrots and daikon, roasted chicken pieces, and lime quarters and chicken fat and juices from the oven pan.
Add the water (or chicken/vegetable stock), pickled lime brine, pickled lime quarters, fish sauce, one tablespoon palm sugar, lemongrass stalks, ginger, a pinch each of salt and pepper, stir, bring to a boil, then turn down the heat and simmer for 40 minutes.
Taste and adjust seasonings if necessary with more fish sauce, salt, or palm sugar: it should be citrus-driven, a little sour, a little sweet, but balanced.
A few minutes before serving, add the cup of roughly chopped fresh coriander and finely chopped scallions or green spring onions, then garnish with a little more fresh coriander and scallions/spring onions at the table.