Preheat oven to 200°C (400°F). Lightly grease muffin tins or ramekins.
Melt butter in a medium saucepan over medium heat. Sauté carrots for 3–4 minutes.
Stir in flour or cornstarch and cook 1 minute. Gradually whisk in chicken broth and milk/cream until thickened.
Season with salt, pepper, and herbs. Remove from heat and stir in chicken, peas, and corn. Cool slightly.
Roll out pastry and cut squares to line muffin tins. Spoon filling into each. Cover with another pastry square or fold edges over. Brush with milk or beaten egg.
Bake 25–30 minutes until golden and bubbly. Cool 5 minutes before serving.