Using a thin, flexible knife, cut chicken breasts away from bones, leaving skin intact.
Place chicken between 2 sheets of plastic wrap; using a mallet or heavy skillet, pound chicken until ½-inch thick (which will help it cook evenly and stay juicy).
Season chicken with salt and pepper. Dust with flour; shake off excess.
Melt butter in a small saucepan. Spoon off milk solids (the white foam that separates from the butter) to make clarified butter. (It has a high smoke point and won't burn over high heat.)
Pour clarified butter into a large ovenproof skillet set over medium-high heat. When butter begins to shimmer, place chicken in skillet skin side down.
Cook, occasionally pressing on chicken with a spatula so skin maintains even contact with skillet, until skin is brown, 5-6 minutes.
Transfer skillet to oven; roast until chicken is just cooked through, 3-4 minutes.
Remove skillet from oven; turn chicken breast over. Let stand in skillet for 1 minute.
Transfer chicken to a plate and let rest for 5 minutes.
Serve with lemon halves alongside.
Recipe by FoodPhreaque at https://www.foodphreaque.com/index.php/2016/09/30/pan-roasted-chicken-paillard/