Add all ingredients for garlic herb butter to food processor bowl. Process until mixture is completely smooth and well-combined. Set aside.
Thoroughly season chicken breasts on all sides with salt and fresh black pepper to taste. Set aside.
Heat large skillet over medium heat. When skillet is hot, add avocado oil and swirl pan to coat completely. Continue warming skillet over medium heat until avocado oil is hot and shimmering.
When oil is hot, add chicken breasts to skillet, working in batches as needed so as not to overcrowd pan. Sear chicken breasts 4 minutes, then gently check underside of chicken for burning. If any burning is occuring, flip chicken breasts over immediately, otherwise cook 1 additional minute, then flip chicken over.
Cook chicken on other side another 4 to 5 minutes, or until chicken is firm and cooked through. Transfer cooked chicken to plate and set aside to rest. Note: if chicken breasts are particularly large/thick and not firm and cooked-through after 8-10 minutes, cover skillet with lid and reduce heat to low. Cook 2 to 3 minutes or until firm and cooked-through.
If skillet has any burnt spots left behind, carefully pour 1 cup water into hot skillet. Let water come to boil, then pour water out and continue with recipe.
Reduce heat under skillet to low. Carefully pour white wine (or chicken broth) into skillet and use spatula or whisk to scrape up any browned bits of chicken that may be stuck to pan. Bring liquid to low boil, then simmer, uncovered, until mixture is reduced by 50%.
Once liquid has reduced to approximately ⅓ cup remaining, add garlic herb butter mixture to skillet. Stir or whisk until butter is completely melted, then return chicken to skillet and spoon sauce over.
Once chicken is well-coated with sauce, immediately remove skillet from heat. Plate chicken breasts with desired sides and additional sauce from skillet if desired, then serve immediately.