Add the flour, yeast, salt, sugar, olive oil and 2/3 cup of warm water (90F) to a standing mixer fitted with a bread hook attachment. Mix on low until ingredients are combined and the mixture resembles like a shaggy mess.
Increase mixer speed to medium and knead the dough for 6 minutes. The dough should be soft and slightly sticky. Allow rising in a warm place for one hour or until doubled in size.
In the meantime preheat your oven to 500F.
Combine cheeses with eggs and grated butter, mix thoroughly and set aside in the refrigerator until ready for use.
Divide the dough in half. Dust your work surface and rolling pin with a little flour and roll each piece into a 12 inch round.
Divide the cheese mixture equally and evenly spread over the dough all the way out to the edge leaving about 1/4 inch seam.
Imagine your dough divided into quadrants. First, bring the top two sections together pinching them closed. Next bring the bottom two parts together, pinching them closed. There should be ample open space in the middle and your bread should resemble the shape of a rowboat. Then fold in the middle edges slightly, about 1 inch, to create a barrier for the cheese.
Move the cheese boats to a sheet tray, brush with a little olive oil or egg-wash and bake for 15 minutes.
After 15 minutes remove open the oven and carefully crack open one egg over the cheese filling. Give the sheet tray a little shake to help the egg settle. Add 3 Tbsp of butter around the egg and place back into the oven. Bake for an additional 5 minutes.