Place a pizza stone or dutch oven or skillet in the oven and preheat to 500°F. Let stone or pan heat for at least one hour.
In a stand mixer fitted with a dough hook, combine yeast, sugar, olive oil, and ⅓ cup warm milk and ⅓ cup warm water.
Allow to sit a few minutes for yeast to start to bubble. Mix on low speed along with flour and salt.
Once incorporated, increase speed and mix until smooth and gluten begins to form, about 4 minutes.
Move the dough to a well-oiled bowl and cover. Let sit in a warm place until the dough has doubled in size. About one hour.
Meanwhile, make the filling. Combine cheeses in a large bowl. Crack eggs and have them ready to go in a separate bowl.
Once dough is ready, turn it out on a lightly floured surface. Divide in half and shape into loose rounds. Cover with a towel and let rest for 15 minutes.
On parchment paper, roll dough into a 10-inch circle. Start rolling in one half of the circle from the edges until you get to the middle. Stop and roll in the other side and pinch the ends, so it looks like a boat shape. Repeat with another dough ball.
Top each with half of the cheese mixture and brush with beaten egg for egg wash.
Bake until cheese melts, about 15 minutes. Remove from oven and make a well in the melted cheese and top with eggs and butter pieces.
Continue cooking for about 5 more minutes. Enjoy hot! Dunking ripped off chunks of the crust into the buttery egg and cheese center.