Heat a nonstick pan on medium heat. Once hot, crumble the turkey sausage into the pan and cook until no longer pink while breaking it up into smaller pieces.
Preheat the oven broiler on low.
Combine olives, shallot, celery, red peppers, capers, carrot, and olive oil in a small bowl. Stir to combine. Season to taste with vinegar, salt, and pepper.
Layer the flatbread with spinach, sausage, vegetable mixture, and cheese. Place on a baking sheet and underneath the broiler until the cheese is melted and the flatbread is golden brown around the edges.