Combine the milk, sugar, and instant coffee in a pot and melt over medium heat (don't bring to a boil).
Add the egg whites to the bowl and use a whisk to beat well. Gradually add the milk, a little at a time, while mixing.
Strain twice (I recommend a very fine sieve).
Pour into individual serving cups (or cocottes or mug cups), removing any bubbles. Cover with aluminum foil and set aside.
Place in a pot filled with boiling water (the water level should be equal to the level of the pudding). Cover with a lid and simmer over extremely low heat for 7-10 minutes.
Shake, and if the puddling is jiggly, turn off the heat and cool as-is for 5 minutes. Once cool, place in the refrigerator to chill, then it's ready to serve.
Another recipe for using egg whites is this affogato-style pudding with a mature taste Recipe ID: 1233596
And here's another egg white pudding, with green tea and brown sugar syrup Recipe ID: 1251323
Recommended for all you chocolate lovers: An egg-white creamy chocolate pudding Recipe ID: 1375872
Story Behind this Recipe
In order to use up leftover egg whites, I made this coffee-flavored pudding that my kids love.
Turn off the heat once the pudding starts to become jiggly; be careful not to over-boil it. Even if it seems soft, it'll stiffen upon cooling. For the milk, use additive-free milk rather than extended-shelf-life, low-fat milk.