Heat the oil in a Dutch oven over medium-high heat.
Season the chicken with ½ teaspoon salt and ¼ teaspoon pepper. Cook until browned, 6 to 7 minutes per side. Transfer to a plate.
Spoon off and discard the fat. Return the pot to medium-high heat. Add the broth and scrape up any brown bits.
Stir in the radishes, carrots, and sugar.
Place the chicken on top of the vegetables and gently simmer, partly covered, until it is cooked through, 15 to 20 minutes. Sprinkle with the chives.