In a large sauté pan, heat 2 tablespoons of oil over medium high heat. Add the pork, break up and cooking until no longer pink. Remove from pan and set aside.
In the same pan, add the additional 2 tablespoons of oil over medium high heat.
Add onions and sauté until golden brown, then add peppers. Continue to cook, stirring often, for about 10 minutes. Once peppers are tender, add vinegar, wine, and sugar. Reduce heat and simmer until liquid is almost all absorbed, about 3–5 minutes. Season with salt and pepper to taste.
Brush flatbread crusts lightly with olive oil and grill, oil side down, until warm and marked with grill marks.
Place flatbreads on a serving board and sprinkle half your cheese overtop. Then add pork mixture. Sprinkle the rest of the cheese over top, then slice and serve.