Preheat grill to 400° to 450° (high) heat. Cut onion into 1/4-inch-thick slices. Brush with 1 Tbsp. olive oil, and sprinkle with 1/4 tsp. salt and 1/4 tsp. pepper. Cut bell peppers into 1-inch-wide strips. Place pepper strips in a large bowl, and drizzle with 1 Tbsp. olive oil. Sprinkle with remaining 1/4 tsp. salt and 1/4 tsp. pepper; toss to coat.
Grill onion and bell pepper strips, covered with grill lid, over 400° to 450° (high) heat 7 to 10 minutes on each side or until lightly charred and tender.
Brush cut sides of rolls with remaining 2 Tbsp. olive oil, and grill, cut sides down, without grill lid, over 400° to 450° (high) heat 1 to 2 minutes or until lightly browned and toasted.
Spread blue cheese on cut sides of roll bottoms; top with arugula, bell pepper strips, chicken, and onion. Spread mayonnaise on cut sides of roll tops. Place roll tops, mayonnaise sides down, on top of onion, pressing lightly.
*French hamburger buns may be substituted. We tested with Publix French Hamburger Buns.
Note: We tested with Saga Classic Soft-Ripened Blue-Veined Cheese.