Poach chicken in water, chop and set aside.
In large skillet, melt butter over medium heat. Add onion, carrot, celery, thyme, salt and pepper. Cook until vegetables are tender, adding the broccoli for the last few minutes.
Sprinkle flour over the vegetables and cook, stirring, for 1 minute. Gradually whisk in milk and chicken broth.
Stirring constantly, cook for 8 to 10 minutes or until mixture begins to simmer and thickens. Stir in chicken and dijon mustard; return to simmer.
Remove from heat and stir in 1½ cups of the shredded cheese.
Preheat oven to 425°F.
Divide filling among four 1-cup ramekins; let cool slightly. Place pastry on lightly-floured surface and cut into four equal squares.
Place one piece of pastry on top of each ramekin, pressing gently to adhere.
Brush egg wash lightly over the pastry. Cut two steam vents in the centre of each pie. Sprinkle the remaining cheese on top of the puff pastry.
Place ramekins on a parchment-lined baking sheet.
If you do not have ramekins - you can use a deep pie dish and leave the pastry as one large piece, be sure to cute many holds in the top to allow steam to escape.
Bake for 20 minutes, or until filling is bubbling and pastry is golden brown.
Let stand for 5-10 minutes before serving as they are VERY hot.