Slice chicken into strips and place into a large ziploc bag. Whisk all marinade ingredients together in a bowl and pour over chicken. Seal bag and move chicken around to coat. Refrigerate until ready to grill.
In a small bowl, stir all ingredients of blue cheese spread together and store in refrigerator until ready to serve.
When you are ready to cook, place all ingredients for the glaze in a small bowl and microwave to melt the butter. Stir and set aside. Preheat outdoor grill to high heat. Remove chicken from refrigerator and grill 2-3 minutes a side or until grill marks form and thoroughly cooked. Spoon glaze over cooked chicken and cover with foil to keep warm.
Place pita on grill to warm for 30 seconds or so.
For assembly, top warm pita with chicken, blue cheese spread, chopped celery, carrot and spinach.