Blot the chicken breasts dry with a paper towel to ensure all the moisture is removed.
Season chicken breasts with salt and pepper, and set aside.
Heat the avocado oil in a large heavy bottom skillet over medium high heat. When the oil just starts to shimmer, add the chicken breasts to the pan. Be sure not to overcrowd the pan, you may have to sear them in batches.
Sear for three minutes per side, until golden brown and internal temperature reaches 155°F. Remove chicken from the pan and set aside.
Reduce heat to medium and add the shallot to the chicken drippings. Move the shallot around the pan and cook for three minutes until it softens. Then, add the minced garlic and cook for an additional minute, until fragrant.
Deglaze pan with wine, being sure to scrape up all the fond on the bottom of the pan. Add the anchovy paste and whisk together.
Add the chicken stock to the pan, and allow it to simmer till it reduces by half and starts to thicken (about seven minutes). Then, add the butter and stir till it melts, and finish with fresh parsley. Add the chicken back into the pan, on top of the sauce. Cover and remove from heat. Let stand for five minutes.
Add the sauce to a plate and serve with sliced chicken.