The Ultimate Vegetarian Club Sandwich

Name: The Ultimate Vegetarian Club Sandwich
Ingredients: 1 Japanese eggplant, cut into 1/4-inch-thick rounds; ⅜ teaspoon kosher salt, divided; ⅜ teaspoon freshly ground black pepper, divided; Cooking spray; 2 tablespoons canola mayonnaise; 1 tablespoon stone-ground mustard; 1 tablespoon prepared horseradish; ⅓ cup pitted kalamata olives plus 1 tsp. olive brine; 8 (1/4-inch-thick) slices seeded whole-grain bread; ½ cup hummus; 1 cup roasted red peppers, sliced; 1 medium ripe avocado, peeled and thinly sliced; 1 Roma tomato, sliced; 2 Persian cucumbers, thinly sliced; 1 to 1 1/2 cups arugula