Pat thighs very dry and season with 1/2 teaspoon salt and pepper. Brush a 12-inch skillet with oil, set chicken skin-side down in the cool pan and place over medium heat. Cook for 12 to 15 minutes until the skin is golden and releases easily.
Turn thighs and cook for 3 minutes, then transfer to a plate while partly cooked. Leave about 1 tablespoon rendered fat in the skillet, spooning off any excess.
Add potatoes, carrots, broth and remaining salt. Cover and cook over medium-low heat for 10 to 12 minutes until vegetables are almost tender.
Uncover and let remaining liquid evaporate for 2 to 3 minutes. Cook vegetables for 4 minutes more, turning once so some faces brown, then stir in garlic for 30 seconds.
Nestle chicken skin-side up over the vegetables with its plate juices. Cook uncovered over medium-low heat for 5 to 8 minutes until chicken reaches 165°F and potatoes are tender.
Transfer chicken briefly to a clean plate. Stir butter, capers and lemon juice through the vegetables, return chicken skin-side up and serve one thigh with one-quarter of the vegetables per person.