Prepare the Oven: Preheat your oven to 375°F, ensuring it’s ready when your dish is assembled.
Cook the Aromatics: In a large, oven-safe skillet, melt the butter over medium heat. Add the chopped shallot, sage, and thyme, stirring frequently until the shallot becomes fragrant and the butter takes on a golden-brown color, about 3 to 5 minutes.
Thicken the Base: Sprinkle the flour into the skillet, stirring constantly, and let it cook until lightly golden, around 1 to 2 minutes. Gradually whisk in the chicken broth and milk until smooth. Add salt and pepper to taste, and bring the mixture to a gentle boil.
Simmer the Filling: Lower the heat to medium-low and let the mixture simmer, whisking occasionally, until it thickens slightly. This should take about 10 minutes. Stir in the chopped carrots and broccoli, cooking for another 5 minutes or until they are just tender.
Add Chicken and Herbs: Remove the skillet from heat and mix in the shredded chicken and parsley. Adjust the seasoning with additional salt and pepper if needed.
Prepare the Puff Pastry: Lightly dust a surface with flour and unfold the thawed puff pastry. If necessary, roll it out slightly so it will fully cover your skillet. Lay the pastry over the skillet, tucking the edges under for a snug fit. Brush the top of the pastry with the beaten egg, and use a sharp knife to cut 2–3 small slits into the top for ventilation.
Bake to Perfection: Place the skillet in the preheated oven and bake for 40 to 45 minutes, or until the puff pastry is a deep golden brown. Remove from the oven and allow it to cool for about 10 minutes before serving.
Serve and Store: Sprinkle the finished dish with whole sage leaves for garnish and serve warm. Any leftovers can be kept in the refrigerator, covered tightly in plastic wrap or an airtight container, for up to 3 days.