Heat a large cast-iron skillet over medium-high heat. Coat pan with cooking spray. Sprinkle both sides of steak with salt and pepper. Add steak to pan; cook 6 minutes. Turn steak; add onion to pan. Lightly coat onion with cooking spray. Cook steak an additional 6 minutes or until desired degree of doneness, stirring onion frequently. Transfer steak to a cutting board. Cook onion for an additional 2 minutes or until tender and lightly browned. Remove pan from heat.
Core pears; cut into thin slices. Place pears in a large bowl. Drizzle pears with 2 tablespoons lemon juice; toss well to coat. Combine remaining 1 tablespoon lemon juice, vinegar, water, olive oil, and garlic in a small bowl; stir well with a whisk. Add salad greens and cheese to pear mixture. Drizzle with oil mixture; toss to coat.
Cut steak diagonally into thin slices. Top each flatbread with about 7 slices (about 2 ounces) steak and about 1 1/3 cups salad mixture.