Mix together the flour, yeast, salt, sugar, olive oil and warm water until it comes together and pulls away from the side of the bowl. You can mix in a stand mixer then mix a couple minutes more to give in effect a knead, but I find it easy enough to mix by hand and then knead around 5 minutes (it's therapeutic too!).
Brush a large bowl with olive oil over bottom and sides. Put the dough in the bowl, turn over to coat and then cover with a cloth. Set aside in a warm place to double in size, around 90 minutes.
Preheat oven to 450F/230C towards the end of the rising time.
Knock back the dough once it has doubled in size then stretch/roll into roughly a circle shape. Line a large baking sheet/tray with parchment and transfer the dough to the parchment and stretch back into a circle.
Mix together the crumbled feta and grated mozzarella.
Put some cheese towards the edge of the dough circle.
Roll in two opposite sides of the dough, including some cheese as you do so, to make an oval. Then twist the ends together, turning over twice to seal up, to form a 'boat'.
Fill the middle of the dough 'boat' with the rest of the cheese.
Bake for around 20 minutes until the bread is lightly brown then carefully add the egg to middle. This is often easier if you break the egg in to a small bowl first.
Carefully return the bread to the oven and cook until the egg white is almost cooked through, around another 5 minutes. Remove from the oven and add a little butter to the egg. Serve warm, breaking up the egg yolk and mixing with the butter. Eat by breaking bits of the bread off and dipping in the egg and cheese filling.