Preheat the oven to 325°F.
Heat a large Dutch oven over medium-high heat. Place the shallots and ginger cut-side down in the dry pot and cook undisturbed for about 5 minutes, until deeply charred. Remove and set aside. Alternatively, char them under the broiler on a baking sheet, turning as needed, or carefully hold them over a gas flame using metal tongs until blackened in spots. Add the neutral oil to the same pot over medium heat. Pat the chicken thighs dry and season generously with kosher salt. Place them skin-side down and sear for 8-10 minutes, until golden brown. Flip and cook for another 2 minutes, then remove from the pot. The oil is optional, as the chicken thighs will render some fat on their own.
Carefully remove the skin from the chicken thighs and spread it out on a lined or nonstick baking sheet. Bake for about 30 minutes, flipping halfway through, until golden and crispy. Set aside on a paper towel to cool.
Meanwhile, return the chicken thighs to the pot and pour in the chicken broth and water, scraping up any browned bits from the bottom. Add the charred shallots, ginger, and smashed garlic cloves. Bring to a gentle boil, then reduce the heat, partially cover, and simmer for about 25 minutes, until the chicken is cooked through and tender.
Remove the chicken from the pot, shred the meat, and discard the bones.
Using a spider strainer or slotted spoon, remove and discard the ginger, shallots, and garlic from the broth.For a clearer broth, I like to strain it through a fine-mesh sieve before returning it to the pot, but this is optional. Add the jasmine rice to the broth and simmer uncovered for 15-20 minutes, stirring occasionally, until the rice is tender and the broth has thickened slightly.
Turn off the heat and stir in the shredded chicken, fish sauce, a squeeze of fresh lime juice, a dash of soy sauce, and a pinch of sugar. Taste and adjust the seasoning as needed.
Ladle into bowls and finish with sliced scallions, crumbled crispy chicken skin, a spoonful of chili oil, and a lime wedge.