Heat the oven to 500°F. In a food processor or coffee grinder, pulse the peanuts just until finely crushed; be careful not to grind them to a paste. Transfer the crushed nuts to a plate or shallow bowl. In a second plate, beat the eggs with the cayenne. Put the flour in a third plate. Line up the flour, eggs, and nuts in that order.
Between two sheets of plastic wrap, lightly pound the chicken breasts to even them out (if you don't have a mallet, use a heavy pan or the side of a cleaver). Lightly season the chicken with salt and pepper. With one hand (this will be your dry hand), dredge a chicken breast in the flour, making sure it's coated evenly. Shake off the excess. Transfer the chicken to the other hand (this will be your wet hand) and dip it in the egg. With the same hand, dredge the chicken breast on one side only in the crushed peanuts, patting to coat the chicken. Set aside, nut side down, and repeat with the three remaining chicken pieces.
Set a large cast-iron skillet over medium-high heat and add just enough oil to make a light film. When the oil is very hot, add the chicken, peanut side down, and cook until the crust is light brown, about 2 minutes. Flip the chicken over, put the skillet in the oven, and let the chicken roast for about 4 minutes. Remove the pan from the oven, check for doneness with the tip of a knife, and serve immediately.
This is delicious with a Burgundy, or a Pinot Noir from Oregon or California.