In an airtight container, combine buttermilk, 1 1/2 t salt and 1/2 t black pepper. Place a chicken breast half between 2 12 in squares of plastic wrap. Using a meat pounder or the bottom of a saucepan, pound chicken until just more than 1/4 inch thick. Repeat with remaining breasts; place in buttermilk marinade. Chill 20 minutes.
Place bread, remaining 1 1/2 t salt, 1 t black pepper, and cayenne pepper in the bowl of a food processor. Pulse until fine crumbs form, about 10 seconds.
Transfer mixture to a shallow bowl.
Remove a chicken breast half from buttermilk mixture, and shake excess liquid.
Dredge both sides of chicken in bread crumbs, and transfer to a plate. Repeat with the remaining chicken breasts.
Line a baking sheet with paper towels. Heat oil in a large nonstick skillet over medium-high heat. When hot, carefully place 2 chicken breast halves in pan. Fry until bottoms are golden, about 2 minutes. Turn chicken breast over, and cook until golden and cooked through, about 2 minutes more. Transfer to prepared pan. Repeat with remaining chicken.