Put the flour, eggs and breadcrumbs in 3 separate shallow bowls. Season the chicken with salt and pepper; dredge each cutlet in the flour, dip in the egg, letting the excess drip off, then dredge in the breadcrumbs until evenly coated. Transfer to a plate.
Heat 1/4 inch of vegetable oil in a large nonstick skillet over medium-high heat. Add the chicken and cook until golden brown and crisp, about 4 minutes. Flip and continue cooking until golden on the other side, about 3 more minutes. Remove the chicken to a rack set on a baking sheet to drain; season with salt.
Meanwhile, heat 1 1/2 tablespoons vegetable oil in another large nonstick skillet over high heat. Add the garlic and ginger; cook, stirring, until golden, about 30 seconds. Add the bok choy; season with salt and cook, stirring, until the bok choy is wilted, about 1 minute. Add the rice and cook, stirring, until warmed through, about 2 minutes. Stir in the scallions; season with salt. Serve the chicken with the rice and katsu sauce.