Begin by placing each chicken breast between two sheets of plastic wrap. Use a meat mallet or rolling pin to gently pound the chicken to an even thickness of about 1/2 inch. This helps the chicken cook evenly. Season both sides with salt and pepper.
Set up a breading station: In one shallow dish, place the flour. In a second dish, beat the eggs until well combined. In a third dish, add the panko breadcrumbs.
Dredge each chicken breast in the flour, shaking off any excess. Then dip it into the beaten eggs, allowing any excess to drip off. Finally, coat the chicken in the panko breadcrumbs, pressing gently to adhere the crumbs.
In a large skillet, heat about 1/2 inch of vegetable oil over medium heat until hot (around 350°F). Carefully add the breaded chicken breasts to the skillet, cooking in batches if necessary to avoid overcrowding.
Fry the chicken for about 4-5 minutes on each side, or until golden brown and cooked through (internal temperature should reach 165°F). Use a meat thermometer for accuracy.
Once cooked, transfer the chicken to a paper towel-lined plate to drain excess oil. Allow it to rest for a few minutes before slicing.
Serve the chicken katsu with tonkatsu sauce drizzled on top or on the side, along with cooked rice and shredded cabbage if desired.