Heat 2 tablespoons avocado oil in large skillet over medium heat, swirling oil around skillet to coat pan evenly. While skillet heats up, generously season both sides of each chicken breast with salt and pepper to taste.
When oil is hot and shimmering, add chicken breasts to skillet. Pan-sear chicken 4 minutes, then flip and add another tablespoon avocado oil. Sear chicken another 4 minutes until cooked-through and no longer pink. Note: if chicken breasts are large and/or not firm after 8 minutes, cover skillet with lid and reduce heat to low. Continue cooking, checking firmness every 2 minutes, until chicken breasts feel springy and firm when gently pressed with tongs.
Transfer fully-cooked chicken breasts to plate and set aside. If chicken charred or left dark spots in skillet, wipe skillet out with paper towel so sauce is not bitter.
Pour ½ cup white wine (or ½ cup chicken broth) into hot skillet to deglaze, scraping up any stuck bits of chicken with spatula. Raise heat under skillet and bring liquid to boil, then add 1 cup chicken broth. Cook liquid until reduced by approximately 50%.
Once liquid has reduced by half, add Boursin cheese pieces. Whisk cheese into liquid until cheese has melted completely.
Return cooked chicken to skillet and spoon sauce over chicken, cooking only until chicken is warmed-through. Garnish with chopped chives and serve warm with plenty of Boursin sauce over mashed potatoes, pasta, mashed cauliflower, or rice.