In a small bowl, whisk together the vinegar, chicken broth, mustard, garlic and a pinch of red pepper, Set aside.
Pat the chicken with a paper towel until dry, then salt generously all over. Set a medium to large oven-proof skillet or pan over medium heat and add olive oil. When the oil is hot, add chicken thighs skin down and cook, untouched, for 9 minutes. When the skins release from the pan, flip the thighs over and cook for 5 more minutes. Remove thighs from pan and set aside.
Add the balsamic mixture to the pan and bring to a simmer, stirring and scraping away the bits on the bottom as you go, 2 to 3 minutes. Meanwhile, set your oven to broil.
Add tomatoes and chicken thighs, skin-side up, to the pan. Spoon the balsamic juices over the tomatoes and the chicken skin, cover the pan and bring to a simmer. Simmer for 3 minutes, spoon more balsamic juices over the chicken thighs, then cover the pan and simmer for 3 more minutes.
Remove the cover, spoon more juices over the chicken thighs, and then place the uncovered pan into the oven to broil until the chicken skin is browned and crispy, 2-4 minutes.
Garnish with fresh basil leaves and serve immediately with crusty bread.