Place the chicken breasts, roughly chopped onion, carrot, celery, peppercorns and bay leaf into a large pan and add enough water to cover the ingredients.
Bring the mixture slowly to the boil, then reduce the heat and simmer for 25-30 minutes, or until the chicken is completely cooked through.
Remove the chicken from the pan and set aside. Keep warm. Strain the cooking liquid and reserve 450ml/16fl oz of it. Set aside.
Heat half of the butter in a large separate pan and add the flour. Stir over a low heat until the mixture forms a smooth paste that leaves the sides and base of the pan cleanly. Gradually add the reserved cooking liquid, stirring well after each addition, to make a sauce. Season, to taste, with salt and freshly ground black pepper.
Add the broccoli and water chestnuts to the sauce.
In a bowl, mix the egg yolk with the cream and slowly add this mixture to the sauce. Add a squeeze of lemon juice and stir to combine.
Cut the cooked chicken into bite-sized pieces and return to the sauce. Stir well.
In a separate pan, heat the remaining butter and fry the remaining thinly sliced onion until soft and golden-brown.
To serve, sprinkle the onion over the casserole, garnish with parsley and serve with the boiled baby carrots.