Pat dry the chicken thighs and drumsticks with a paper towel. This will help give the chicken a good sear as well as help the salt and pepper stick to it. Season chicken with salt and pepper on both sides.
Melt butter in a large skillet over medium-low heat. Brown chicken on both sides, leaving it to cook for 5 minutes on each side. Remove the chicken pieces from the skillet and transfer them onto a plate to rest.