Pat the chicken thighs dry with paper towels, and season both sides with salt and pepper.
In a large mixing bowl, combine balsamic vinegar, chicken broth, tomato paste, lemon juice (reserve zest for later), honey, minced garlic, and 1 tablespoon minced basil. Add seasoned chicken thighs, toss to coat with balsamic mixture, and set aside, or, if time allows, cover mixture and refrigerate for up to 4 hours.
Place a 12-inch skillet over medium heat, and add chicken and marinade. When the liquid comes to a boil, cover, reduce heat to medium-low, and simmer 5 to 7 minutes. Turn chicken, increase heat to medium and cook, uncovered, until chicken is no longer pink at the center and juices run clear, 5 to 7 minutes more. An instant-read thermometer inserted near the center of chicken should read at least 165 degrees F (74 degrees C).
Remove chicken and any accumulated juices from the skillet to a serving platter. Tent with foil to keep warm.
Wipe the skillet clean and over medium-high heat, add olive oil. When the oil is hot, carefully add tomatoes to the skillet. Cook undisturbed for 2 minutes, allowing the tomatoes to blister. Stir gently, and continue cooking for an additional 2 minutes. Sprinkle tomatoes with a pinch of salt.
Remove foil tent, pour tomatoes over chicken, and top with shredded mozzarella. Sprinkle reserved lemon zest and torn basil over all. Serve warm.