Preheat the oven to 425 degrees. Line a large baking sheet with parchment paper or foil.

In a large bowl, combine all three cheeses and mix well to combine. Set aside.

Cut the pizza dough into 2 equal pieces. Roll each piece into a 9- to 10-inch circle.

Place the cheese mixture in the center of each piece of dough, about 2 inches from the edges. Roll up two sides of the dough towards the center. Pinch or twist the ends together, creating a canoe or boat shape. Transfer the boats to the prepared pan.

Brush the edges of the dough with the beaten egg.

Bake for 15 to 20 minutes, until the crust is golden brown. Remove the pan from the oven, and keep the oven at 425 degrees.

Use a small spoon to make 5 indents in the cheese mixture of each bread, about 1/2-inch deep. Crack the quail eggs into the indents. Sprinkle the diced butter over the cheese, 1 tablespoon of butter per bread.

Return the pan to the oven and bake for 5 to 7 more minutes, until the egg white is cooked (the yolks can/should still be runny). Note that the eggs will continue to cook as they sit in the hot cheese mixture.

Top with chives, if using, and serve immediately.