Gently mix chicken, mayonnaise, 1/4 cup panko, salt, and pepper. Shape four patties about 4 inches wide and 1/2 inch thick; refrigerate on a lined tray 15 minutes.
Mix celery, yogurt, vinegar, and blue cheese; refrigerate while cooking. Put flour, beaten egg, and remaining 3/4 cup panko in three shallow bowls.
Lightly coat chilled patties in flour, egg, then panko, gently pressing crumbs. Heat oil in a large skillet over medium until a loose crumb steadily sizzles.
Fry in two batches 4 to 5 minutes per side until golden and centers reach 165°F. Adjust heat to prevent burning; briefly drain on a rack.
Melt butter with hot sauce in a small saucepan over low. Brush cooked patties on both sides, dividing sauce evenly.
Broil cut bagel faces 1 to 2 minutes. Top each bottom with one patty and an even layer of celery slaw; close.
Rest 1 minute before cutting; serve while crisp.