1. Preheat grill to medium-high heat.
2. Slice eggplant into 1-inch-thick slices; brush both sides evenly with 1 tablespoon oil. Place eggplant in a single layer on a grill rack coated with cooking spray; grill 6 minutes on each side or until eggplant is tender. Brush both sides of bread slices evenly with 2 tablespoons oil. Place bread slices in a single layer on a grill rack coated with cooking spray; grill 1 minute on each side or until toasted.
3. Combine eggplant, 1/4 teaspoon salt, 1 tablespoon juice, and next 3 ingredients (through garlic) in a food processor; pulse until coarsely chopped. Spoon about 1 heaping tablespoon eggplant mixture on each bread slice.
4. Combine arugula, tomatoes, and mint in a bowl. Drizzle with remaining 1 1/2 tablespoons juice and remaining 1 tablespoon oil. Sprinkle with remaining 1/4 teaspoon salt; toss to coat. Divide salad mixture evenly among bread slices; top evenly with cheese.
Wine note: Bourillon-D'orleans Vouvray Sec, Vieilles Vignes, Coulee d'Argent 2008 ($22), is a bright, crisp, medium-bodied chenin blanc from the Touraine region of the Loire Valley. Its racy backbone of lemongrass and straw has enough body and acid to stand up to the grill flavors and cut the creamy tartness of the yogurt. --Alexander Spacher