While the vegetables continue to roast, pat the chicken dry with paper towels; season with salt and pepper on both sides. In the same pan, heat 2 teaspoons of olive oil on medium-high until hot. Add the seasoned chicken, skin side down. Cook, occasionally pressing down with a spatula, 5 to 6 minutes, or until the skin is browned and crispy. Flip and cook 4 to 5 minutes, or until browned and cooked through. Leaving any browned bits (or fond) in the pan, transfer to a plate. Set aside in a warm place. Carefully drain off and discard any excess oil from the pan.
Add the vinegar and 1/4 cup of water to the pan of reserved fond (be careful, as the liquid may splatter). Cook on medium-high, stirring constantly and scraping up any fond from the bottom of the pan, 30 seconds to 1 minute, or until thoroughly combined. Add the fig spread and remaining butter. Cook, stirring constantly, 30 seconds to 1 minute, or until the butter has melted and the sauce is slightly thickened. Turn off the heat; season with salt and pepper to taste.
Divide the roasted vegetables and cooked chicken between 2 dishes. Top the chicken with the fig pan sauce. Top the vegetables with the brown butter chestnuts. Enjoy!