Pat the chicken dry with paper towels. Poke a small opening in the clear membrane between the skin and the flesh of the chicken breast. Glide your index finger through the opening, loosening the skin from the flesh, without removing it completely. You're creating a big pocket in which the butter will sit! Repeat with the other chicken breasts.
Put a spoonful of the butter mixture through the opening you made in the skin of the chicken breast. Once it's in there, smooth out the butter by gliding your finger over the skin, until it's evenly distributed. Repeat with the other breasts. Arrange the chicken breasts on the sheet tray lined with the cooling rack. Wash your hands thoroughly.
If you wish, drizzle the skin with a little oil, for extra crispy skin. Season with salt, to taste. Put the potatoes and the chicken into the oven, and roast, turning the pans once halfway through the cooking time and stirring the potatoes, about 30 minutes. Relax, sip a glass of wine, and bask in the crackling sounds of butter and comforting smell of the roasted chicken!
Chicken is ready when an instant-read thermometer inserted into the thickest part of the breast, but not touching the bone, registers about 160 degrees F. Remove the pans from the oven and tent with foil. Allow them to rest about 5 minutes. Transfer the chicken and potatoes to a serving platter and serve.