In a serving bowl, whisk together red wine vinegar, Dijon mustard, 2 1/2 teaspoons oregano, 2 teaspoons olive oil, and 1/4 teaspoon pepper; set aside.
Place steamer basket in saucepan filled with a few inches of water; cook edamame, covered, until tender (about 3 minutes). Transfer edamame to a bowl. Add string beans to steamer; cook, covered, until tender (about 2 minutes). Add beans to edamame. Add tomatoes and olives; toss to combine.
Heat a lightly oiled grill pan over medium-high heat. Brush 1 teaspoon oil on one side of pitas; grill, turning, until golden (about 2 minutes). Transfer pitas to a plate. Brush 1/2 teaspoon oil evenly on one side of cheese slices; sprinkle with the remaining oregano and pepper. Grill cheese, seasoned side down, until marks form (about 1 minute); transfer to a plate.
Place 1 pita half on each of 4 plates; top with bean salad and cheese. Drizzle with remaining olive oil.
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