Mix together mayonnaise and Sriracha in a small bowl; set aside.
To make the brine, in a small, shallow bowl whisk together the the water, salt, and sugar until the solids are dissolved. Pound each chicken breast to a 1/2 inch thickness. Place the chicken breast slices in the brine and refrigerator for 30-60 minutes.
While the chicken is brining, assemble the rub by mixing together paprika, garlic powder, onion powder, thyme, oregano, cumin, salt, black pepper, and cayenne pepper in a small bowl.
Remove chicken from brine and pat dry. Brush the chicken slices lightly with olive oil, then sprinkle evenly with the spice mixture. Flip the chicken over and repeat.
Clean and oil the grate. Brush onion slices lightly with olive oil on both sides. Season with salt and pepper. Grill onions and pepper, turning occasionally, until peppers are completely charred all over and onions until charred on both side and slightly softened, 5-8 minutes total. Remove from grill and place pepper in a paper or Ziploc bag and seal. When the pepper is cool enough to handle, about 5 minutes, remove skin and slice into 1/8-inch strips. Remove skewer from onion and separate into individual rings.
Place chicken slices on grill and cook over high heat until first side is blackened, 3 to 5 minutes. Flip and place slice of cheese over chicken and grill until second side is blackened and chicken is cooked through, 3 to 5 minutes longer. Remove from grill and allow to rest for 5 minutes. While the chicken is resting, place rolls on the grill, cut side down, and grill until lightly browned. Remove from grill, spread top side of bread with Sriracha mayonnaise and assemble sandwich with the chicken, onion rings, pepper slices, and lettuce. Serve immediately.