
Heat 2 tablespoons of the oil in a large nonstick skillet over medium heat. Add the onions; cover and cook, stirring occasionally, for about 5 minutes or until translucent. Reduce the heat to low; add the apple, vinegar, and sugar. Continue to cook, stirring occasionally, until the onions are very tender and nicely browned, about 15 minutes. Add salt and pepper to taste. Remove from the heat and set aside.
Just before serving the pizzas, preheat the broiler. Place the pita halves, rough-side up, on a baking sheet. Lightly brush with olive oil. Broil 4 to 5 inches from the heat source for about 1 minute or until lightly toasted.
To assemble the pizzas, spread the crusts with the shredded cheese and top with the onion mixture. Sprinkle with the walnuts. Broil for about 1 ½ to 2 minutes, or until the cheese is melted.
To serve, cut each pizza into 8 wedges and serve immediately.