Preheat oven to 350°F. Pat chicken thighs dry with a paper towel and season skin with salt and pepper.
In a 10" or 12" cast iron skillet, heat olive oil and butter over medium heat.
Once butter is melted, increase stove temperature to medium high and after 1-2 minutes, add chicken thighs, skin side down. Season non-skin side with salt and pepper.
Brown the chicken thighs, skin side first for 5-6 minutes until the skin is golden brown and releases easily from the skillet. Flip and brown the opposite side for 4 minutes.
Transfer chicken thighs onto a sheet pan and roast in 350°F oven for 20-25 minutes, or until the internal temperature reaches 165°F. Allow chicken to rest for 10 minutes before serving.
Apple Cider Pan Sauce
After you transfer the chicken thighs onto the sheet pan and place them into the oven, discard remaining fat in skillet. Be sure to leave the brown bits on the bottom of the skillet - they add the most flavor to the sauce!
Heat 1 tbsp. olive oil in skillet set to medium heat.
Add shallots to the skillet and cook for 2-3 minutes, until translucent.
Add minced garlic and cook for 1 minute. Increase the burner to medium high heat.
Deglaze the skillet with apple cider, bringing up the brown bits stuck to the bottom, using a wooden spoon.
Add chicken stock and apple cider vinegar. Bring to a boil and then reduce to a simmer. Simmer until sauce reduces by half.
Whisk in dijon mustard, tarragon, and salt and pepper. Whisk in butter and remove from heat once butter is melted.
Drizzle sauce on chicken thighs right before serving. Don't do this too soon, because the sauce could make the crispy chicken skin soften. Enjoy!