Prepare the chicken breasts:

Slice the chicken breasts in half horizontally to create 4 chicken cutlets. Place the cutlets between two large pieces of parchment paper and pound each lightly to 1/4-inch thickness. Transfer to a gallon-size zip-top bag.
Make the marinade:

Thinly slice the green onions, setting the dark green parts aside for garnishing. Add the white and light green parts to a small food processor or blender along with the buttermilk and garlic. Blend until smooth. Reserve 3 tablespoons of the marinade if you’d like to make a sauce for serving.
Transfer the remaining marinade to the bag with the chicken. Seal the bag and turn to coat the chicken. Place it in the refrigerator and marinate for at least 30 minutes and up to 8 hours.
Cook the chicken:

Set a large cast iron skillet over medium heat and add the oil. Once hot, remove the chicken from the marinade, shaking off any excess, and add to the skillet. Discard the chicken marinade.
Season the tops with salt and pepper and cook until golden brown, about 5 minutes. Flip and cook until golden brown on the other side and the internal temperature reaches 165°F, 2 to 4 minutes. Transfer the chicken to a cutting board and let it rest for 5 minutes.
Make the sauce (optional) and serve:

If using, add the mayonnaise to the reserved marinade, season with salt and pepper, and stir to combine. Serve the chicken garnished with sauce and the sliced green onions.