I don't really respond to threads asking how to get to an early breakfast because I don't get up that early, so I don't know anything about it firsthand. But I understand the parks run buses for early breakfasts.
If you can get a bus, the thing to do here is ride to MK on the bus and take the resort monorail back around to GF. (It's also possible to take the small boat, which has its first stop at GF and could be faster, but I don't have any idea if the boats run early. I think the resort monorail does.)
At normal times I allow at least 45 minutes to an hour for any Disney transportation that isn't direct (that is, anything that goes between resorts). At that time of day I might lean toward the hour, I don't know.
If you are pressed for time, the best way is probably to take a taxi.