Prepare the chicken breast by slicing in half, and with a meat mallet, pound until it is very thin, around ¼ inch. Salt and pepper on both sides. Add the flour to a bowl and coat the chicken on both sides.
In a stainless steel pan over medium heat, add avocado oil. Once hot, pan sear the chicken, working in batches to not overcrowd the pan. Sear 3 minutes per side, or until golden brown. Remove from the pan and set aside.
In the same pan over medium heat, add 1 tbsp of butter. Add the shallots and sliced lemon and saute until translucent, salting as you go, around 3-5 minutes. Add minced garlic and saute another 1 minute. Add the chicken stock, scraping up any brown bits off the bottom of the pan, and bring up to a simmer. Reduce by half.
Once the stock is thickened, turn the heat to low and stir in the capers and remaining cold butter to emulsify and make creamy. Once it is incorporated and thickened, serve over the chicken and top with parsley.
