In a small bowl, whisk together the vinegar, chicken stock, mustard, garlic, and a pinch of red pepper. Set aside.

Pat chicken thighs with a paper towel until the skin is dry, then salt generously all over. Set a medium to large oven-proof skillet or pan over medium heat and add olive oil. When oil is hot, add chicken thighs skin-down and cook, untouched, for about 9 minutes. When the skins release from the pan, flip the thighs over and cook for 5 minutes more. 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. 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 covered for about 6 minutes.

Remove the cover, spoon more balsamic 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 to 4 minutes.

Garnish with fresh basil leaves and serve immediately over rice, pasta, or polenta, or with a few chunks of crusty bread.