With the rack in the middle position, preheat the oven to 180 °C (350 °F).
In a saucepan, soften the onion, celery, carrots, and green beans in the butter for about 8 minutes over medium heat.
Add the chicken and cook over high heat until lightly browned. Season with salt and pepper. Reduce the heat to medium. Add the garlic and cook for 1 minute.
Sprinkle with the flour and stir to combine. Add the milk and bring to a boil, stirring with a whisk. Simmer gently for 2 to 3 minutes.
Add the remaining vegetables and parsley and stir well. Adjust the seasoning. Pour into a 20-cm (8-inch) square baking dish. Set aside.
On a baking sheet, spread the strips of phyllo dough and drizzle with the butter. Toss gently. Scatter lightly over the vegetable mixture, without pressing. Bake for about 30 minutes or until the pastry is golden brown.
As you only need very little phyllo dough, thaw the roll partially. Cut the amount of dough you need and put the roll back in the freezer, well wrapped.
Recipes with the "Healthy Pick" stamp have been evaluated by a registered dietitian member of the Ordre professionnel des diététistes du Québec.