In a large bowl, combine the flour, sugar, salt, and instant yeast. Add the warm water and vegetable oil, then mix until the dough comes together.
Knead the dough on a floured surface for about 8-10 minutes until smooth and elastic.
Place the dough in a greased bowl, cover with a clean kitchen towel, and let it rise for 1 hour, or until it doubles in size.
In a medium bowl, combine mozzarella, feta, and Monterey Jack cheese. Mix in 1 egg to bind the cheese mixture. Set aside.
Preheat your oven to 450°F (230°C). Line a baking sheet with parchment paper.
Punch down the risen dough and divide it into two equal portions. Roll each portion into an oval shape, about 10 inches long.
Spoon half of the cheese filling onto the center of each oval, leaving a 1-inch border around the edges. Roll the edges inward, pinching the ends to form a boat shape.
Transfer the shaped breads onto the prepared baking sheet. Bake for 15 minutes, or until the bread is golden brown and the cheese is bubbling.
Remove the breads from the oven and use a spoon to make a small well in the center of the cheese. Crack one egg into the well of each bread.
Return to the oven and bake for another 4-5 minutes, or until the egg whites are set but the yolks remain runny. Top each bread with 1 tablespoon of butter while still hot. Serve immediately.