Heat a 5-quart pot over medium-high heat. Add the oil and garlic and cook, stirring often, until the garlic is fragrant and golden, about 2 minutes. Use a slotted spoon to transfer the garlic to a paper-towel lined plate, and reserve for serving. Reserve the garlic-infused oil in the pot, off the heat.
Season the chicken with 1/2 teaspoon salt and 1/4 teaspoon pepper. Return the pot to the stove over medium-high heat, then add the chicken, skin-side-down and cook until the skin is golden, turning halfway through, about 8 minutes. Transfer the chicken to a bowl. Add the onion, ginger, and rice to the pot. Cook, stirring often, 1 to 2 minutes. Add the chicken broth, 4 cups water, a pinch of saffron, and the bay leaf to the rice. Return the chicken and juices to the pot. Bring to a boil, then reduce the heat to maintain a vigorous simmer.
Cook, partially covered, until the chicken is tender, the rice is cooked, and the soup is thickened, about 30 minutes. Discard the bay leaf, stir in the fish sauce and lime juice, and season the soup with salt and pepper. Serve one chicken thigh per bowl of soup and garnish with the fried garlic, scallion greens, freshly ground pepper, reserved saffron threads, and lime wedges.
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