Heat a large, heavy bottom skillet or casserole pan over medium high. Use a paper towel to pat the skin dry on the chicken thighs, season on both sides with kosher salt and freshly ground pepper.
Working in two batches, place the chicken thighs skin side down and cook for 7-10 minutes, until the skin is golden brown, flip over and cook for another 5-7 minutes. Set the chicken aside while you make the cherry tomato sauce.
In the same pan with the remaining chicken fat, add the tomatoes, vinegar, calabrian chili paste and kosher salt.
Stir occasionally, letting the tomatoes burst and release their juices, 8-18 minutes.
Add the chicken thighs back into the pan and reduce heat to medium. Cook for another 7-10 minutes, until the chicken thighs are fully cooked and reach an internal temperature of 165 degrees F. Top with goat cheese and fresh basil leaves and serve immediately.
Grape tomatoes tend to take longer to burst and release their juices. If you are using them, it will probably take closer to 20 minutes to get them to the stage when you add the chicken thighs back to the pan.
If you like it spicier, feel free to add more chili paste.
You can find Calabrian chili paste in Italian markets and on Amazon. You can also substitute with a pinch or two of crushed red pepper flakes if you prefer.
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