2 tablespoons vegetable oil
2½ pounds skin-on, bone-in chicken legs and thighs
8 garlic cloves, peeled
⅓ cup (packed) light brown sugar
¼ cup (or more) unseasoned rice vinegar
2 slices ¼"-thick slices peeled ginger
1 cup low-sodium chicken broth
¼ cup reduced-sodium soy sauce
2 scallions, thinly sliced
Cooked white rice (for serving)
Heat oil in a large wide heavy pot over medium-high heat. Season chicken with salt and, working in 2 batches, cook until golden brown and crisp, 6–8 minutes per side; transfer to a plate. Add garlic to pot and cook, stirring often, until golden, about 2 minutes; transfer to plate with chicken. Pour off fat from pot.
Return pot to medium-high heat and add ½ cup water, scraping up browned bits. Add brown sugar; stir to dissolve, then cook, stirring, until mixture thickens and turns a deep amber color, about 4 minutes. Carefully add vinegar (it may bubble up; sugar will crystallize); stir to dissolve sugar.
Add ginger, broth, and soy sauce, then add chicken, skin side up, and garlic. Bring to a boil, reduce heat, and simmer gently until chicken is cooked through, 20–25 minutes. Transfer chicken to a plate.
Bring cooking liquid to a boil and cook until thick enough to coat a spoon, about 10 minutes. Return chicken to pot; turn to coat. Top with scallions and serve with rice.