Prepare the Chicken: If your chicken breasts are very thick, you can pound them gently to an even thickness of about 1/2 inch. This ensures even cooking. Season both sides generously with salt and pepper.
Set Up Breading Station: Prepare three shallow dishes. In the first, place the all-purpose flour. In the second, whisk the eggs until smooth. In the third, place the panko breadcrumbs.
Bread the Chicken: Dredge each chicken cutlet first in the flour, shaking off any excess. Then, dip it into the beaten eggs, ensuring it's fully coated. Finally, press the chicken firmly into the panko breadcrumbs, making sure the entire surface is covered. Repeat for both chicken breasts.
Heat the Oil: Pour vegetable oil into a heavy-bottomed pan or Dutch oven to a depth of about 2-3 inches. Heat the oil over medium-high heat until it reaches 350°F (175°C). You can test the oil by dropping a few panko crumbs in; they should sizzle immediately.
Fry the Katsu: Carefully place one or two breaded chicken cutlets into the hot oil, being careful not to overcrowd the pan. Fry for about 3-5 minutes per side, or until golden brown and cooked through. The internal temperature should reach 165°F (74°C).
Drain and Rest: Using tongs, remove the chicken katsu from the oil and place it on a wire rack set over a baking sheet to drain any excess oil. Let it rest for a few minutes before slicing.
Serve: Slice the chicken katsu into strips. Serve immediately with steamed rice, shredded cabbage, and a generous drizzle of tonkatsu sauce.