In a medium saucepan, bring the vinegar, broth, honey and tomato paste to a boil, stirring well. Simmer the vinegar sauce until reduced to ½ cup, about 8 minutes.
Heat 2 tablespoons of olive oil in a large, heavy skillet. Season the chicken breasts with salt and pepper; add half of them to the skillet. Cook over moderate heat, turning once, until browned. Transfer to a plate. Repeat with remaining breasts.
Add remaining tablespoon of olive oil to skillet. Add garlic and shallots, and cook over low heat for 5 minutes. Add the wine; boil until reduced to ¼ cup. Add the vinegar sauce and bring to a simmer.
Return the chicken to the skillet. Cover and simmer over low heat for 10 minutes; turn over and cook another 10 minutes. Transfer to serving plates and spoon sauce over.
Adapted from Food and Wine’s Chicken in Red Wine Vinegar recipe; here, I’ve made the dish lighter by using olive oil and boneless chicken breast, and omitting the crème fraiche. The fig balsamic vinegar gives this dish a fruity, tangy flavor and a rich dark color.