Preheat the oven to 400°F (200°C).
Heat olive oil in a large skillet over medium heat.
Add the chopped onion and garlic, sauté for 3-4 minutes until softened.
Add diced carrots, celery, and mushrooms to the skillet and cook for 5 minutes until vegetables begin to soften.
Stir in the frozen peas and cook for 2 minutes.
Sprinkle the whole wheat flour over the vegetables and stir well to coat evenly.
Gradually pour in the chicken broth while stirring continuously to avoid lumps.
Add the skim milk, dried thyme, and dried rosemary. Stir and cook until the mixture thickens, about 5 minutes.
Season with salt and black pepper to taste.
Remove the skillet from heat and gently fold in the shredded cooked chicken.
Transfer the chicken and vegetable mixture into a 9-inch pie dish.
Place the whole wheat puff pastry sheet over the filling, trimming any excess pastry around the edges.
Press the edges to seal and cut a few small slits on the top to allow steam to escape.
Brush the pastry with beaten egg white for a golden finish.
Bake in the preheated oven for 25-30 minutes or until the pastry is golden brown and crisp.
Remove from the oven and let the pie rest for 5 minutes before serving.