In a screw-top jar combine the olive oil, vinegar, lime juice, thyme, salt, and ground red pepper. Cover and shake well. Place greens in a large bowl; toss with dressing, chopped tomatoes, and tomatillos. Arrange greens on 8 plates. Set aside.
Using half of the shredded cheese, divide evenly over 4 of the tortillas; sprinkle lightly with cumin. Layer with tomato slices, shrimp, poblano pepper, and onion. Top with the remaining cheese and tortillas.
Spray nonstick coating in heavy skillet or griddle; place over medium heat. Cook each quesadilla 2 minutes or until golden. Turn carefully. Cook 2 minutes more or until cheese melts and bottom is golden. Repeat with remaining quesadillas.
Cut each cooked quesadilla into 6 wedges. Arrange 3 wedges on top of each plate of greens. Makes 8 main-dish servings.