My Big Fat Greco Inspired Burger

Name: My Big Fat Greco Inspired Burger
Ingredients: 1 small cucumber; 1 1/2 cups plain yogurt; 1 clove garlic, minced; 3 tablespoons fresh dill; 1/2 cup crumbled Greek feta cheese; 3 tablespoons white wine vinegar; 1/2 teaspoon oregano; 1/2 teaspoon salt; 1/4 teaspoon ground black pepper; 1/4 cup olive oil; 1 large eggplant; 2 pounds ground lamb; 3/4 cup chopped fresh mint leaves; 1 1/2 tablespoons ground cumin; Salt and pepper; Oil, for brushing on the grill rack; 6 small pita breads; 3 cups chopped romaine lettuce