Trim excess fat from chicken parts and cut breasts in half so that all the pieces will cook in roughly the same amount of time. Season the chicken generously with salt and pepper. Heat vegetable oil in a large, oven-proof pot (such as a Dutch oven) over medium-high heat. Brown chicken on all sides, about 5 minutes per side, then set aside on a plate and reserve. Add the onions and garlic and cook a couple minutes to sweat them down a little. Add the ale and deglaze the bottom of the pot. Add the tomato paste and canned tomatoes and stir well until tomato paste is dissolved. Add carrots, thyme, and plums. Return the chicken to the pot along with any reserved juices. Gently submerge the chicken pieces by nestling them into liquid (this will cook them evenly without drying them out).
Return to medium heat and bring to a simmer. Reduce heat to low, cover, and continue to simmer on low for 1 1/2 hours until chicken is cooked through and starting to fall off the bone.