Prepare the chicken. Butterfly the chicken breast by slicing breasts in half horizontally, creating thin slices of chicken. Or, cut the chicken breast in half vertically. Place between saran wrap or inside of a plastic bag and flatten with a meat mallet to desired thickness (about ¼ inch). Either method will produce thin slices of chicken that will cook evenly and quickly.
Sprinkle chicken with salt and pepper on both sides. Place flour in a large shallow dish. Dredge chicken in flour and set aside on a plate.
Heat 1 tablespoon of avocado oil in a large frying pan over medium-high heat. Let the pan and oil heat up before adding the chicken. Once the pan is hot, add half of the chicken to the pan. Only add half to avoid overcrowding. Cook about 3 minutes, or until browned, on one side. Flip over and cook another 3 minutes, or until browned on other side. Transfer chicken from the pan to a plate and cover with foil. Add remaining 1 tablespoon of avocado oil to the pan and repeat with other half of chicken.
In the same frying pan, melt the butter and heat the garlic until fragrant. Add the wine and chicken broth. Bring sauce to a boil, scraping up any brown bits left from browning the chicken. Reduce to a simmer and cook about 1 to 2 minutes, or until sauce is slightly reduced. Mix in lemon juice and capers. Add chicken back to the pan, coating generously with lemon-caper sauce. Cook another 2 to 3 minutes, or until sauce thickens. Remove from heat. Top with parsley.