Mix together the flour with the salt, black pepper, and cayenne and place on a plate. Whisk together the eggs with the buttermilk. Lightly sprinkle the breasts with salt and pepper then dredge each into the flour. Dip the flour-coated breasts into the eggs and then dredge in the flour again. Place the breaded chicken breasts on a sheet pan.
Heat up the oven to 200°F. In a large heavy skillet, such as a cast-iron skillet, on medium-high heat up an inch of oil to 350°F, about 5 minutes. If you don’t have a thermometer, you can test the temperature by sticking a wooden spoon into the oil. If it bubbles around the spoon, it should be ready for frying.
Working in batches, gently lower each breast into the oil and cook for 2 minutes per side, or until lightly browned, turning once. Drain on a paper towel and place in the oven while you fry the remaining breasts.