It might help to freeze the egg patty separately from the rest of the sandwich.
This might give you a chance to brush off any ice crystals that form, meaning less water to try and deal with inside the sandwich.
You could either assemble the sandwich when you make it in the morning (with the bonus of being able to heat the elements separately, and possibly discard more water that way), or else after the eggs are frozen, brush off any ice crystals, assemble the sandwich around the frozen egg and put in the freezer again to make your frozen convenience sandwich.
Usually I see more ice crystals when something's been in the freezer longer, one reason why making the sandwiches on a delay might help more (maybe make the egg one one week in advance, assemble the whole sandwiches to freeze with last week's frozen egg?) - but it does work that removing the crystals, brushing the ice off, does dry out the item a bit more.
Also, this might help the egg freeze more quickly - as per Joe's answer - so the trapped water freezes more quickly, does less damage to the egg, and stays in the egg instead of dripping out. The smaller thermal mass means it should cool and freeze a lot quicker than a whole sandwich, where the bread in particular may insulate it for a longer freezing time.
And it might make it easier, working with a sturdy and already frozen egg patty, to dust the egg in cornstarch or wrap it in rice paper or something, if you want another layer to absorb moisture with.