Preheat oven to 375 degrees F. Arrange pita bread pieces on a 15x10x1-inch baking pan. In a bowl whisk together olive oil, thyme, sesame seeds, 1/4 teaspoon salt, and 1/4 teaspoon ground black pepper. Drizzle mixture over pita pieces; toss to coat. Spread in an even layer. Bake 10 to 12 minutes or until golden and crisp, stirring once.
On a large platter arrange greens, mint, parsley, melon, cucumber, and red onion. Sprinkle with cheese and pomegranate seeds. Top with pita chips. Pour Lemon Vinaigrette over all; toss to coat.
In a screw-top jar combine olive oil, pomegranate juice, red wine vinegar, lemon juice, lemon peel, and honey. Cover; shake well.