Gently mix ground chicken with mayonnaise, 1/4 cup panko, salt and pepper. Shape into four patties about 4 inches wide and 1/2 inch thick and refrigerate on a lined tray for 15 minutes.
Mix shredded celery with yogurt, vinegar and blue cheese and refrigerate while cooking. Put flour, beaten egg and remaining 3/4 cup panko in three shallow bowls.
Coat each chilled patty lightly in flour, then egg, then panko, pressing crumbs gently to adhere. Heat oil in a large skillet over medium heat until a loose crumb sizzles steadily.
Fry patties in two batches for 4 to 5 minutes per side until golden and 165°F in the center, adjusting heat so the crust doesn’t burn. Drain briefly on a rack.
Melt butter with hot sauce in a small saucepan over low heat. Brush the cooked patties on both sides with the sauce, dividing it evenly.
Toast bagel cut faces under the broiler for 1 to 2 minutes. Put one Buffalo chicken patty on each bottom, add an even layer of celery slaw and close with the tops.
Rest for 1 minute before cutting and serve while the coating is still crisp.