Step 1

Season eggplant with 1/4 teaspoon salt and 1/4 teaspoon pepper. Generously coat with cooking spray. Heat a cast-iron grill pan over medium-high. Working in batches, add eggplant, cut sides down, to pan, and grill 2 to 3 minutes, turning 1 or 2 times. Transfer to a plate; let cool.

Step 2

Stir together mayonnaise, mustard, and horseradish in a small bowl. Set aside. Using a small food processor, whirl the olives and brine into a spreadable puree.

Step 3

Grill bread directly over a medium flame, turning often, about 1 minute to slightly char the surface and impart a bit of smokiness. Let cool briefly.

Step 4

Spread about 2 tablespoons hummus on each of 4 bread slices, and top evenly with eggplant, olive puree, roasted red pepper, avocado, tomato, cucumber, and arugula. Spread 1 1/2 teaspoons mayonnaise mixture on each of the remaining 4 bread slices. Sprinkle evenly with remaining 1/8 teaspoon salt and remaining 1/8 teaspoon pepper, and top the sandwiches. If you like, add a little more arugula for bite. Serve sandwiches whole or cut in half.