1. Combine first 3 ingredients in a small saucepan; bring to a boil. Remove from heat, and add jalapeño; let stand 15 minutes. Drain.
2. Preheat grill to medium-high heat.
3. Place whole eggplants on grill rack; grill 15 minutes or until charred on all sides and tender when pierced with a thin knife, turning frequently. Remove from heat; gently scrape off charred skin, if desired. Cool eggplants slightly; halve lengthwise. Lightly score flesh.
4. Combine yogurt, tahini, 1/4 teaspoon salt, and garlic, stirring well. Drizzle 4 teaspoons oil and lemon juice over eggplant flesh; top with yogurt mixture.
5. Combine arugula, remaining 2 teaspoons olive oil, and remaining 1/4 teaspoon salt in a medium bowl; toss gently. Top each eggplant half with about 1/2 cup arugula, 5 jalapeño slices, 1 tablespoon pomegranate arils, and 1 tablespoon pecans.