Whenever a recipe says "Au Bain Marie" you may feel you are on the right track to become a chef. But what is it exactly and why do we need this complicated method? Or can we bypass it with some simpler method?
Au Bain Marie first of all raises a question: who was Marie? The method is supposedly named after Mary of Alexandria, Egypt, who was an alchemist. Whether or not she was successful in creating gold we don't know, but we did inherit this method of preparation.
The method works like this: have a pan of water, bring it to a slow boil, and hang a bowl in the pan, without the bowl touching the water. Practically: do not fill the pan too high, and make sure your bowl is wide enough to rest on the sides of the pan.
As the water boils, the steam will rise up and touch the bowl. This will heat the bowl slowly, but never to more than 100° Celsius (212° Fahrenheit), the boiling point of water (more on the boiling temperature of water here). So we now have a well-controlled cooking process: the bottom of the bowl has a maximum temperature, the air above is a little above room temperature, and buy stirring enough (or whisking if you want to emulsify or aerate) you make sure the whole mixture gradually warms up.
You will have to stir and keep your eye on the mixture at all times. Make sure you know which temperature is "enough" for your preparation. For an egg yoke to solidify, you will need a temperature of 65 - 70° Celsius (149 - 158° Fahrenheit), way below the boiling point of water.
Au Bain Marie is the preferred method to deal with egg mixtures that should not split or turn into scrambled egg at a certain temperature, such as a sabayon or a sauce Hollandaise.
By the way, there also is a cold water Au Bain Marie method, to slowly cool down sauces et cetera.
Can you replace Au Bain Marie with other methods? Yes, you can as long as you control the temperature. Here is how James Martin does it: by clarifying the butter and add it to the egg mixture once the butter has cooled down some.
It is a simple matter of math: roughly the sum of the weight of the butter times its temperature + the weight of the egg yolk mixture times its temperature divided by the weight of butter and mixture combined should be in the 70° Celsius zone.
So, (Wbutter * Tbutter + Wegg * Tegg) /(Wbutter + Wegg) = 70 (all temperatures in Celsius).
Assume 300 gr of clarified butter at 90°, 3 egg yolks of 18 gr each and 6 grams of gastric both at 30° would bring the mixture to 80°. As there is much more butter than egg mixture, the temperature of the butter will be the deciding factor.
Disclaimer: the above does not take the specific heat property of butter and egg mixture into account. Please feel free to adjust the above formula for this refinement!