Prepare the marinade: In a deep plate or shallow dish, combine 1/2 cup smoked tomato dressing, 1 garlic clove (minced), 2 tablespoons fresh thyme leaves, 1/4 teaspoon salt, and 1/2 teaspoon pepper.
Marinate the chicken: Add the 4 boneless, skinless chicken breasts to the dish and turn to coat. Cover and let sit for 30 minutes.
Preheat the grill: Heat an outdoor grill or indoor grill pan to medium-high.
Grill the chicken: Place the marinated breasts on the hot grill and cook about 10 minutes per side, or until fully cooked through and juices run clear (internal temperature 165°F/74°C).
Rest and slice: Transfer the cooked chicken to foil, tent and rest 5 minutes, then slice into approximately 1/4-inch thick strips.
Assemble sandwiches: Spread 2 tablespoons Sriracha Caesar dressing on the bottom half of each ciabatta roll. Top with one slice of cheese, some arugula, sliced red onion, about 1 tablespoon kimchi slaw, chicken slices, sun-dried tomatoes, and another slice of cheese.
Press and melt: Place each sandwich in a panini press or sandwich press and press for about 4 minutes, until the cheese melts and the sandwich is warmed through.
Serve: Cut sandwiches as desired and serve with fries, fresh tomatoes, and extra smoked tomato and Sriracha Caesar dressings for dipping if desired.