In a large skillet over medium-high heat, melt 2 tablespoons of butter. Sauté the chopped carrots, onion, and red bell pepper for about 5 minutes, stirring occasionally. Incorporate the mushrooms along with ½ teaspoon of salt and ¼ teaspoon of black pepper. Continue cooking for an additional 5 minutes until the mushrooms are lightly browned. Add minced garlic and thyme; cook for 1–2 minutes until aromatic. Stir in the peas and set the skillet aside.

In a separate saucepan set over medium-low heat, melt the remaining 4 tablespoons of butter. Whisk in the flour and cook continuously for 1 minute. Pour in the white wine, cooking for another 1–2 minutes until nearly evaporated. Gradually whisk in the heavy cream and chicken broth. Increase the heat to medium-high, bring to a boil, then reduce to low. Let it simmer until thickened, stirring occasionally. Fold in the reserved vegetables and the shredded chicken. Season with the remaining ½ teaspoon of salt, ¼ teaspoon of black pepper, and the chopped parsley. Remove from heat.

Preheat your oven to 400°F (200°C). Evenly distribute the chicken mixture among six individual ramekins.

Lightly flour the puff pastry sheets and roll them out to a thickness of ¼ inch. Position a ramekin on the pastry sheet and cut out a square approximately ½ inch larger on all sides. Alternatively, cut pastry rounds matching the ramekin diameter. Drape each ramekin with a pastry square or round, pressing gently around the edges to seal. Use a sharp knife to cut three small slits in the top of each pastry to allow steam to escape. Brush the tops with the egg wash.

Place the ramekins in the oven and bake for 30 minutes or until the puff pastry is deeply golden and the filling is bubbling. If browning occurs too quickly, loosely tent with foil. Serve hot, directly from the ramekins or transferred to individual plates.