Crab Salad-Stuffed Pitas

1 tablespoon(s) extra-virgin olive oil

1 teaspoon(s) lime juice Freshly ground pepper , to taste

1 teaspoon(s) minced fresh ginger

7 ounce(s) cooked crab meat , drained (see Note)

1/2 cup(s) finely chopped celery heart with leaves

2 tablespoon(s) minced red onion

1 tablespoon(s) chopped fresh cilantro

1 teaspoon(s) minced jalapeño pepper , or to taste

2 6-inch pita breads , warmed and cut in half crosswise

2 large lettuce leaves , torn in half