Season chicken with salt and pepper in a shallow bowl; generously coat with panko bread crumbs, pressing crumbs into chicken.
Heat peanut oil in a large skillet over medium-high heat; fry chicken until bottom is browned and crispy, 4 to 5 minutes. Flip each chicken piece over; top 4 of the pieces with green onions and 1 Cheddar cheese slice. Fry chicken until bottom is golden brown and is no longer pink in the center, 4 to 5 more minutes.
Coat both sides of hamburger rolls with mayonnaise. Place 1 cheese-cover chicken piece and 1 plain chicken piece on 1 roll bottom; place the top of roll over the chicken.
