Put the egg whites into the bain-marie and egg yolks into a separate bowl (ideally a jug).
Whisk the egg yolks with three-quarters of a teaspoon of sieved cornflour. Stir in the lemon juice and zest. Set aside
In the top of the bain-marie, (heat off) whisk the egg whites with a pinch of salt until light and frothy.
Turn the bain-marie to medium-low heat and slowly drizzle in the egg yolk mixture. Whisk gently until the sauce starts to thicken. About 2-3 minutes. You can now slowly add the stock until you reach the desired consistency. Start with one tablespoon of stock. The avgolemono sauce needs to cook for another 5-6 minutes. Add more stock once its thickened to get to your desired consistency.
Use the stock from whatever dish you add the lemon sauce to, chicken or vegetable water or add a cup of premade stock of your choice, which has been boiled and then allowed to cool slightly. You can add seasoning; pinch white pepper, a generous grinding of black pepper, pinch finely ground saffron, or chopped herbs. Dill and fennel work pair very well with chicken or asparagus but also think about lemon thyme or tarragon?
Notes
This method may appear a little fussy as many recipes make the Avgolemono sauce all in one pan, but if you overheat it, it may split and taste too eggy because it is overcooked. It will also lose its unique airy character; The sauce is light complementing and not dominating the accompanying dish.