Place each chicken breast between two pieces of parchment or wax paper. Using a meat mallet or the bottom of a heavy skillet, pound the cutlets until they are 1/2-inch-thick. Season the chicken with salt.
Add the flour to one large, shallow bowl. Whisk the eggs in a second large, shallow bowl. Add the Panko breadcrumbs to a third large, shallow bowl.
Dredge each of the chicken breasts in the flour, shaking off any excess, then dip it into the eggs and immediately into the Panko, pressing the breadcrumbs to adhere to all sides of the chicken.
When ready to cook, line a baking sheet with paper towels. Place a large skillet over medium heat. Add enough oil to generously coat the bottom of the pan. Once the oil is hot, add the breaded chicken breasts (careful not to overcrowd the pan and working in batches, if needed) and cook them, undisturbed, until they are golden brown. Flip the chicken once and continue cooking until it reaches an internal temperature of 165°F.
Transfer the chicken Katsu onto the prepared baking sheet and immediately season them with salt.