In a bowl, whisk together the ice water, 2.5 tablespoons salt, and 2.5 tablespoons sugar until fully dissolved. This brine will keep the chicken moist and add flavor from the inside out. Cut the chicken breasts in half horizontally to create thinner, more evenly-cooked pieces, then submerge them completely in the brine. Let them sit for 30 minutes at room temperature—this brining time is crucial for tender, juicy chicken.
While the chicken brines, combine all the dry spices in a small bowl: paprika, garlic powder, onion powder, thyme, oregano, cumin, 0.5 teaspoon salt, black pepper, cayenne pepper, and brown sugar. Whisk them together thoroughly to ensure even distribution. I like to make this spice blend ahead of time so it's ready the moment the chicken finishes brining—it saves precious time when you're ready to cook.
Seed and slice the red bell pepper into 1/4-inch strips. Slice the red onion into thin 1/8-inch rings and place them in a small bowl. Drizzle the onions with a bit of olive oil and season lightly with salt and pepper—this helps them caramelize evenly on the grill. Have the lettuce washed and ready for assembly, and split and set aside the club rolls for toasting later.
Remove the brined chicken from the liquid and pat it completely dry with paper towels—this is essential for achieving a proper blackened crust. Coat each piece lightly with 2.5 tablespoons olive oil, then generously coat both sides with the spice blend from Step 2, pressing it gently so it adheres well. Preheat your grill to medium-high heat for about 5 minutes until it's hot and the grates are clean. I prefer to oil and season the chicken right before grilling so the spices don't absorb moisture and lose their punch.
Place the seasoned peppers and onions from Step 3 directly on the hot grill grates. Let them cook for 5-8 minutes, stirring occasionally, until they're charred and softened. Once the vegetables are nearly done, place the spiced chicken pieces on the grill alongside them. Cook the chicken for 3-5 minutes until the bottom is deeply blackened, then carefully flip each piece. Immediately top each piece with a slice of pepper jack cheese and cook for another 3-5 minutes until the chicken is cooked through and the cheese is melted.
Remove the chicken from the grill and let it rest for 3 minutes—this allows the juices to redistribute, keeping the meat tender and moist. While the chicken rests, place the split club rolls on the grill cut-side down for about 1-2 minutes until they're lightly golden and toasted. Spread a generous amount of sriracha mayonnaise on both sides of each toasted roll, preparing them for assembly.
On each prepared roll, layer a piece of the rested chicken with melted cheese, followed by the charred peppers and onions from Step 5, and finish with a handful of fresh green leaf lettuce. Close the sandwich with the top half of the roll and serve immediately while everything is still warm. The combination of the smoky blackened chicken, creamy sriracha mayo, charred vegetables, and cool crisp lettuce creates a perfectly balanced sandwich.