Heat 1 Tbsp. vegetable oil in a large Dutch oven over medium. Add chicken skin side down and cook, lifting pieces with tongs once or twice toward the end to let hot fat flow underneath, until fat is rendered and skin is crisp and golden brown, 7–10 minutes. Transfer skin side up to a plate. Because drumsticks are covered entirely in skin, you can brown both sides, but it’s not necessary.
Add ¾ cup vinegar, ¾ cup soy sauce, 1 Tbsp. brown sugar, 5 bay leaves, and reserved garlic and sliced chile to Dutch oven. Bring to a simmer, then reduce heat to low, stirring to dissolve sugar. Season generously with pepper.
Return chicken to pot skin side up. Cover and cook chicken very gently, adjusting heat to maintain a bare simmer and turning pieces once, until meat is very tender and pulling away from the bone (but not so tender that it’s falling apart), 35–40 minutes.
While chicken is simmering, cook 1 cup rice in a medium pot according to package directions. Transfer chicken to a clean plate. Increase heat to medium-high and boil braising liquid, shaking pot often, until liquid has formed a rich sauce thick enough to coat the back of a spoon, about 5 minutes. Remove pot from heat and return chicken to pot, turning to coat in sauce.
Serve chicken and sauce over rice. Thinly slice remaining chile and scatter over, then season with more pepper.