Step 1Clean and quarter the potatoes. Place them in a bowl of cold water and set aside.
Step 2Wash the chicken under cold running water and pat dry with paper towels. Remove excess fat and season each piece with a pinch each of salt and pepper.
Step 3Heat the oil in a large heavy skillet over medium-high heat until hot. Quickly sear the chicken thighs until browned on both sides, 2 to 3 minutes per side; taking care as the chicken will splatter and adjusting the heat as necessary. Remove the browned chicken to a plate.
Step 4Drain the potatoes and pat dry with paper towels. Add them to the hot skillet, cut side down, and cook until lightly browned, 3 to 4 minutes. Remove them to a plate and cover to keep warm.
Step 5Add the shallot and capers to the skillet and quickly saute, about 1 minute. Stir in the chicken broth, vinegar and lemon juice, scraping up any browned bits in the bottom of the pan.
Step 6Add the chicken back to the pan, and bring to a gentle simmer. Cover with a tight-fitting lid. Cook the chicken until tender and a thermometer inserted into the meat reads 165 degrees, about 25 to 30 minutes. Add the browned potatoes to the pan during the last 15 minutes of cooking.
Step 7Arrange the chicken and potatoes on a platter. Taste the sauce to check the seasoning, then bring to a simmer and cook for an additional minute. Spoon the sauce over the chicken, and sprinkle with the chives. Serve immediately.