Whisk together olive brine and buttermilk. Put the turkey breast into a resealable plastic bag and pour brine-buttermilk mixture into the bag. Add rosemary and thyme sprigs. Seal bag and refrigerate for 8 hours.

Take the bag out of the fridge and allow to rest until the breast reaches room temperature.

Preheat an air fryer to 350 degrees F (175 degrees C).

Cook the breast in the air fryer for 15 minutes. Flip over the breast and cook for 5 minutes until turkey breast is no longer pink in the center and the juices run clear. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). If internal temperature is lower, keep cooking at 5 minute intervals until the correct temperature is reached.