Line a plate or sheet pan with 2 layers of paper towels. Pour oil into a large cast iron skillet until you have roughly ¼ inch of oil in the skillet. If you have an exhaust in your kitchen, turn that on. Heat oil over medium-high heat. You can tell when the oil is hot enough by dropping in a breadcrumb and seeing if it sizzles. Use a fork or metal spatula or tongs to gently place chicken into the pan, in a single layer. (You will fry the chicken in multiple batches. Don’t try to fit it in all at once.) Cook until the bottom is a golden brown, about 3 minutes. Then flip over and cook until the second side is browned, about 3 more minutes. Once the second side has cooked, remove chicken from pan and place on paper towel-lined plate. Use a spider to remove any breadcrumb pieces that might have fallen off from the oil. (Otherwise they may get stuck to the chicken in your next batch and burn.) Continue until all chicken is cooked. Add a layer of paper towels to your plate between each layer of chicken to soak up excess oil. (If you want this step to go faster, you could heat oil in a second skillet to fry 2 batches at once. I did 3 batches in a single skillet.)
Troubleshooting: If your chicken is not browning in 3 minutes, the oil is not hot enough. If you are running out of oil in the pan because the chicken soaked up a lot, your oil is not hot enough. Turn the heat up a notch and wait for oil to get hotter before doing your next batch of chicken. Alternatively, if the breading gets very dark brown or burns, lower heat and wait a few minutes for oil to cool down before adding more chicken. If all this is making you nervous, try frying just 1 small piece of chicken cutlet first as a test.