Preheat oven to 350°F Season chicken inside and out with salt and pepper. In a large ovensafe saucepan or Dutch oven, heat oil on medium-high. Add chicken and brown, breast side down, 5-10 minutes. Turn over and brown second side. Transfer chicken to a platter. Add celery and onion and sauté 5 minutes. Add carrots and sauté another 5 minutes. Remove pot from heat.
Place chicken, breast side down, overtop vegetables . Combine broth, water and vinegar and pour over chicken. Add potatoes to the pot, cover and braise 45 minutes. Turn chicken, breasts side up, cook another 10-15 minutes with lid off to brown.
Remove chicken and vegetables from pan and keep warm. Skim fat from cooking liquid and discard. Carve chicken and place slices into shallow bowls, adding vegetables and broth. Garnish with fresh chives and serve.