As the method you are describing is actually steaming rather than boiling:
There are two components in the question that are relevant to the answer: volume and sticking
For the volume, what you actually need is a container that will allow you to cook the egg white evenly in a relatively thin layer of about 2-3 cm (1 inch) depth, so as to increase the surface area:volume ratio. To do this you need a large flat plate shaped container (similar shape to the one in the link) rather than a deeper pot. This allows much more exposure of the egg to the heat from the steam and subsequently lowers cooking time and evenness. You need to be able to suspend this over the water, so you need a large saucepan or wok with corresponding lid to hold the plate you are using and trap the steam so that the egg can cook. You also need a trivet to raise the plate with the egg out of the water, where it can steam rather than boil in the water. Something like the one in this link is best, in my opinion.
With regards to the sticking - there's not a whole lot that you can do about this other than use a non-stick pan or a seasoned iron dish. You can use a thin layer of oil (if you see it running you've used too much) over the surface of the dish to aid in stopping sticking, but I have found this to have varying success.