How to soft boil eggs with a countertop egg steamer?
I happen to have some experience with steam egg cooker.

For starters: how does an egg cooker work?

The egg cooker doesn't use a timer. Instead, the cooking time is determined by the amount of water you put into it: the more water, the more time it takes to boil off and trigger the overheat alarm. This is why your egg cooker has a water measure cup that has markings for different number of eggs to cook.

Secondly, to soft boil an egg you don't actually control the absolute temperature, but control the temperature gradient: you want the outer layers of the egg to be very hot (rolling boil water or steam) but the inner core of the egg doesn't really have time to heat up, which is why 6min in rolling boil water gives you reliably good results, although you may need to adjust a fraction of a minute depending on the temperature of your fridge. Strong temperature gradient (difference between outer and inner parts of the egg) is what gives you the "water balloon effect" e.g. the yolk is near liquid and white is like a thick and strong membrane.

OK then how do you steam your egg?

For your situation you basically need to reverse engineer your measure cup that comes with the steamer. I propose following procedure:


Put about 1-2 tbsp of water in your steamer, turn it on, wait till
boiling. 
Put egg in, steam for 6-7 mins.
Turn steamer off and check one egg, three things can happen:

a) This egg is under cooked, you can salvage them by cooking them for another minute.

b) This egg is over cooked, then your data is useless, you need to wait for another chance to repeat this experiment and deal with 4 medium done eggs for now.

c) The egg is perfectly cooked.
If you hit case a) or c) you are in luck, wait for the steamer to cool and measure the remaining water. Subtracted from your initial water you get your desired water amount for 4 eggs.


Based on my experience, 1 tbsp and 7 min should yield a close enough result and I'm sure you can get this right within 3 iterations.

How do I know all this?

I once owned a egg steamer (6 eggs) and lost the water measure cup as well. I ended up putting about 2tbsp of water and cook for 7 min. It takes roughly 20sec for water to boil and generate steam and 40s for the surface of the egg to equalize with steam, so the net cooking time is about 6min and result is mostly perfect. At the end there's always some excessive water, but I never bothered to reverse engineer the measure cup and stuck to the 7min rule. In fact, because I never carefully measured water consistently, the net cooking time varied and fluctuates the doneness of the end product. If you control the water and time constantly you will get much more consistent results than I did.