Sprinkle both sides of each eggplant slice with salt. Place eggplant slices on paper towels; let stand 20 minutes. Blot dry with paper towels.
Combine mayonnaise, mustard, rosemary, lemon juice, and garlic in a medium bowl. Remove 1/4 cup mayonnaise mixture; set aside. Add arugula to mayonnaise mixture in bowl; toss gently.
Coat both sides of the eggplant slices with cooking spray. Place on grill rack, and grill 5 minutes on each side or until lightly browned. Transfer eggplant to a plate. Coat both sides of tomato slices with cooking spray. Place tomato slices and bread slices on grill rack; grill 2 minutes on each side or until tomato is tender and bread is lightly browned.
Place 1 bread slice on each of 4 plates. Top each slice with 1/3 cup arugula mixture, 2 eggplant slices, 2 tomato slices, and 1 tablespoon goat cheese. Drizzle 1 tablespoon reserved mayonnaise mixture over each serving. Top with remaining bread slices.