Preheat a grill pan over medium-high heat. In a medium saucepan, heat the EVOO, 2 turns of the pan, over medium-high heat. Add the onion and garlic; cook until softened, about 5 minutes. Add the tomatoes and chicken stock and heat through, about 5 minutes. Stir in the vinegar, brown sugar, Worcestershire sauce and lots of pepper. Lower the heat and simmer, stirring occasionally, until thickened, 7 to 8 minutes.
Drizzle the chicken with EVOO and season with salt and pepper. Cook on the grill pan, turning once, for 6 minutes. Remove from the pan and layer 1 chicken piece with 2 mozzarella slices, another chicken piece and another mozzarella slice. Repeat 3 more times to make 4 double-decker chicken stacks. Tent with foil for 1 minute to soften the mozzarella cheese.
Combine the arugula and basil. Pile one-quarter of the greens on each of the 4 bottom pieces of ciabatta. Pile the chicken stacks on top, then cover with the barbecue sauce and bread tops.