Heat oven to 250ºF (121ºC) with rack set in lowest position. Heat oil in a large (7- or 8-quart) Dutch oven over medium heat. Pat chicken dry then season all over with salt, about 2 teaspoons total, rubbing some directly on breast meat underneath skin.
Carefully add chicken, breast side down, to pot with hot oil. Add onions and bay leaves, nestling them around chicken. Cook chicken until breast is deep golden brown, 5–6 minutes. Using tongs, flip chicken breast side up and brown until bottom side is deep golden brown, 5–6 minutes.
Meanwhile, add ⅓ cup adobo sauce to 1 ¾ cups water, stir until combined; pour over chicken. Brush remaining tablespoon adobo sauce over breast and legs of chicken. Bring liquid in pot to a simmer then cover with lid and transfer to oven.
Roast chicken until meat can easily be pulled from bone and an instant-read thermometer inserted in the breast register 165ºF (74ºC) and legs register 172ºF (78ºC), 1½–1¾ hours.
Using tongs, carefully transfer chicken to a cutting board (it may fall apart—that’s OK). Skim off and discard any grease from surface of broth. Simmer broth over medium-high heat until reduced and thickened, 20 minutes. Season sauce to taste with salt (I start with ½ teaspoon).
Meanwhile, remove and discard chicken skin. Remove meat from bones and shred into small pieces; add back to pot with reduced sauce and toss to combine.
Served shredded chicken with lime wedges (a squeeze of fresh lime juice adds a lot here) tortillas and desired toppings.