Preheat oven to 425F. Peel carrots and slice into 1/2" thick rounds and set aside. Pat split chicken breasts dry and season on top and bottom with coarse sea salt and cracked black pepper.

Heat oil in a large cast iron skillet or other oven safe skillet over med-high heat. When the oil is warm add the breasts to the pan skin side down to sear. Sear until the skin is crispy and brown about 5 minutes, then flip the breasts and cook on the other side for 3 minutes.

Remove the skillet from the heat and the chicken from the skillet. Add the cut carrots into the skillet with cubed butter on top. Sprinkle thyme on the carrots and add the chicken back into the pan skin side up.

Place skillet in the oven to bake at 425F for 25-27 minutes. Use a meat thermometer to check that the internal temperature of the chicken breast has reached 165F. Remove from the oven and allow to cool slightly before slicing.