I did the following for the above which served 2 people. Comes out really soft and fluffy, and not eggy at all.
4 slices white toast bread (or whatever soft and fresh thick bread you have).
3 eggs
1/4 cup milk
splash of vanilla
cinnamon
Cut the crusts off slices of bread and cut lengthwise in half.
Break 3 eggs into a bowl. Spice up with a little cinnamon. Put in a few drops of vanilla. Add about a 1/4 cup of milk. Mix up with a fork.
Heat up a pan to a medium heat. One at a time, put a piece of bread into the egg mixture, then flip over with tongs. You are really only dipping/immersing them quickly for 5 or 6 seconds total, not soaking for any longer than that. Pick up, let excess drip off for a second or two, then put into pan. Cook each side for about a minute tops, until golden and just starting to tinge brown. When you flip one over, get another one started so you can get into a rhythm and get through them quickly.
The key is probably not soaking the bread in the egg too much, and not having your pan too hot. If you find it is still too eggy for your liking, water down with more milk.