How can I address watery eggs in a microwaved breakfast sandwich?
I see two possible problems with what you've described --


If you don't freeze the egg quickly enough, you're more likely to have larger ice crystals form, resulting in a puddle of water no matter how you defrost it.  You should be able to test for this case by just putting the frozen egg in a covered bowl and let it come to room temperature.  If this gives you a lot of water, you can freeze the eggs on a sheet pan lined with waxed paper or parchment, then assemble the sandwiches to freeze.
Overheating eggs will cause the proteins to tighten resulting in puddles of moisture.  (this is mainly a problem with scrambled eggs or egg casseroles).  The solution to this is to heat it as little as possible.


As for the bread soaking through -- you can also look for alternatives.  Bagels and crusty rolls hold up much better to moisture, but bagels in particular can end up really chewy after microwaving.  A thicker roll would also have more crumb to absorb moisture.

You can also try spreading something on the bread to prevent the moisture from soaking it (although, it can make for messier eating if the sandwich is dripping).  For egg sandwiches, I like hot pepper jellies.