Mix onions, water, lime or lemon juice, the 2 tablespoons olive oil, jalapeno, garlic, thyme, salt, and black pepper in a resealable plastic bag set in a shallow dish. Add chicken. Seal bag and marinate chicken and onions in the refrigerator for 2 hours, turning the bag occasionally.
Drain chicken, reserving marinade.
Cook chicken, half at a time, in the 2 tablespoons hot oil in a Dutch oven until lightly browned, turning to brown evenly. Remove chicken; set aside. Repeat with remaining chicken. Drain fat. Return all chicken to Dutch oven.
Add reserved marinade mixture and chicken broth. Bring to boiling; reduce heat. Cover and simmer for 35 to 40 minutes or until chicken is tender and no longer pink.
Transfer chicken to a serving bowl; keep warm.
Measure juices; skim off fat. If necessary, add enough water to juices to equal 2 cups. Return the juices to the Dutch oven.
Stir together the cornstarch and 2 tablespoons cold water in a small bowl. Stir cornstarch mixture into pan. Cook and stir over medium heat until thickened and bubbly. Cook and stir for 2 minutes more. Pour sauce over chicken. Serve over hot cooked couscous or rice.