4 skinless, boneless chicken breast halves
3 tablespoons olive oil
1 large shallots, peeled and quartered
1 cup chopped green onions
1 1/2 cups chicken broth
1 cup chicken meat, cut into 2 inch pieces
1 1/2 teaspoons salt
1/2 teaspoon ground black pepper
1 1/2 cups shredded Cheddar cheese
1/2 pound shredded Monterey Jack cheese
Place chicken breasts in a saucepan with water to cover; place over medium heat. Bring to a full rolling boil until they are a large, about 15 minutes.
Place olive oil in a heavy skillet you've prepared in advance. Add shallots and green onions. Saute briefly, then add the chicken and green onions. Season with chicken broth, and stir. Season with salt, black pepper and Cheddar cheese. Scrape oven or broiler pan and set aside.
While chicken is cooking, return chicken breasts to a large skillet and pour in 3 tablespoons of chicken broth.
Heat oil in a large skillet over medium-high heat. Stir in 2 tablespoons chicken and green onion, and cook, stirring occasionally, for approximately 4 to 5 minutes. Reduce heat to medium, and add Monterey Jack cheese. Return chicken to skillet. Simmer, stirring occasionally, for about 5 minutes (the longer they cook, the more they stick together). Fluff breasts with fork while stirring in the cheese. Reduce heat to low. Simmer for 30 minutes, stirring constantly. Reserve 2 tablespoons broth if the serving dish is still warm and can be used with another dish.
Remove pita from trimmings. Grab 2 tablespoons of cheese from the center of each and place on bottom of trimmings.