Cook the chicken in salted water with the bay leaf until tender. Let it cool, then shred the meat. Strain and keep the stock.
In a large pan, melt the butter and sauté the onion until soft and slightly golden. Add the carrots and cook for a few minutes until tender.
Stir in the flour and cook for about a minute. Slowly add the chicken stock while stirring to avoid lumps. Add the milk or cream and let it simmer until thick and smooth.
Add the shredded chicken, peas, salt, pepper, thyme, and mixed herbs. Taste and adjust seasoning. If needed, add a little lemon juice for balance. The filling should be creamy but not runny. Let it cool slightly.
Preheat the oven to 200°C.
Line a pie dish with one sheet of puff pastry. Spoon in the cooled chicken filling.
Cover with the second sheet of pastry. Seal the edges and make a small slit in the top for steam.
Brush with beaten egg.
Bake for 30 to 35 minutes until golden brown and puffed.