Start a fire in your cooker with mesquite charcoal. Set the grid in its normal position and allow the cooker to heat to 500 degrees.
Mix the balsamic vinegar, dijon mustard and capers. Set aside until you are ready to serve.
Cut the romaine lettuce and savoy cabbage lengthwise into quarters. If either is very large, cut them into 6 pieces. When the fire is ready, brush the cabbage and romaine with olive oil, and sprinkle with kosher salt and ground black pepper.
Place the cabbage on the grill. Grill for about 90 seconds, turning every 15 seconds. Add the romaine and continue grilling both for an additional 90 seconds, again turning every 15 seconds. You are looking for some wilting and charring. Move to serving platter and add quartered tomatoes. Serve warm with dressing.