For dough, add yeast to the two cups of water for several minutes until it becomes foamy; stir in sugar, olive oil and vinegar.
While yeast is dissolving in water, add the flour and salt to food processor jar and pulse a couple times.
Add the water and yeast mixture gradually to the flour mixture and process until the dough begins to form a ball and come away from the sides of the container. Turn out onto floured surface. Sprinkle with flour and knead for several minutes until the dough is smooth and not sticky. Place in oiled bowl, cover with towel and let sit for two hours until double in size.
After two hours, turn out onto lightly floured surface, knead a couple times and cut in 1/2 or in three equal pieces. Form into balls and use as needed. To use the next day, rub lightly with olive oil, cover with plastic and refrigerate overnight (I prefer using the next day for better flavor).
To make the pizza, preheat oven (with pizza stone if you have one) to 500 degrees F. Place rack in center of oven.
Partially cook the bacon in a pan (don't let it get crisp), remove and set aside on paper towels.
Crack eggs into separate cups (this makes it easy to add them to the pizza when it's in the oven).
Stretch out your dough to desired shape and size. Dust pizza peel (or parchment paper) with a little cornmeal and lay your dough on top, reshaping and leaving a little more thickness on the edges.
Evenly arrange cheese and tomatoes on dough. Then add the bacon and basil (see image). Lightly rub a tiny bit of olive oil on the edges.
Slide onto pizza stone and let cook for about 5 minutes. Carefully add the eggs, one at a time, to the center of the pizza and cook for about another 5 minutes until egg is at desired doneness and the dough is nicely browned on the edges.
Remove and garnish with the fresh parsely.
Prepare to feast on a delicous breakfast!