Another way to think about this is that you are sculpting the bread by removing part of its crust and insides so that, in the end, it looks like a boat, or a pizza crust. How deep the bottom should be, and how wide the edges (where you keep the top crust) is your decision: I recommend leaving 1/4 of the original height on the bottom, and a 1 inch wide edge. Also form undercuts on the edges, by slicing between the remaining parts of the top and bottom crusts, but be careful to keep the crust of the boat intact: you want it to hold in its interior all the goodies you'll place there (melted cheese and eggs yolks).
In case Ciabatta sounds Greek to you, it is a type of bread; you can use instead
Next, grate
That's the hardest ingredient to find; you'll have to look at groceries that import Mediterranean food. If you have a hard time finding it, you can use instead
Put the grated cheese inside the sculpted loaf.
Fry
Layer them on top of the cheese, side by side.
Also fry only partially
Layer them on top of the bacon, one near each end of the loaf, placing the sunny side down; later, we'll stick the loaf in the oven, and we want the yolk to remain runny, so don't cook the eggs fully: just enough to make them hold their shape.
Then melt
Brush the crust of the loaf with half of it. Bake in moderate oven (350 degrees) for about 30 minutes, or until the cheese has fully melted but the bread hasn't burned and the egg yolk is still runny. While the loaf is baking, brush the crust with the remaining butter; if you have any leftover butter, pour it over the eggs.
Serve as soon as it comes out of the oven, cutting the loaf in half. Instruct your guests to break the egg on their half so that the yolk spreads over the melted cheese.
You can think of this dish as a Greek pizza without tomato sauce. The reason the above is a cheater recipe is that we didn't bake our own bread... but it's much easier this way. Also, feel free to substitute any kind of topping you like instead of bacon and eggs: ham, tomatoes, or even steak, brisket, or souvlaki will do just fine. Thus you can also compose vegetarian alternatives.