Step 1

Whisk 1 teaspoon oil, mint, cumin, and pinch of freshly ground black pepper in medium bowl to blend. Add cheese slices; toss to coat. Marinate at room temperature at least 30 minutes and up to 2 hours. DO AHEAD: Can be made 2 days ahead. Cover; chill.

Step 2

Prepare barbecue (medium-high heat). Grill pitas until warm and grill marks appear, about 1 minute per side; transfer to serving plate. Grill cheese until grill marks appear and cheese begins to melt, about 1 minute per side. Stack each warm pita with slice of tomato and cheese. Drizzle with oil and sprinkle tomatoes lightly with sea salt.

Step 3

Available at some supermarkets and at specialty foods stores, natural foods stores, and Greek markets.