Place a large pot of salted water over a high heat and allow it to come to a boil. Scrub any dirt off of the potatoes and then cut them into 2.5 cm (1 inch) cubes. When the water boils, add the potatoes and parboil them for 5-7 minutes. They should still be fairly firm. Drain the potatoes into a colander and toss them a couple times to chuff up the edges and make them slightly fluffy. Set aside.
Place a large non-stick skillet over medium-low heat and add the prosciutto slices. Cook the slices for 1-2 minutes on both sides until crisp. Remove from the pan and set aside to crisp up.
Turn the heat up to medium and add the olive oil to the pan and then the sliced onion. Cook the onion until it just begins to soften, about 4 minutes, and then add the garlic and saute until fragrant.
Pile in the potato cubes and the pesto and season with salt and pepper to taste. Toss everything well so it’s covered in the pesto and then spread it out into an even layer. Allow the potatoes to fry for about 5-6 minutes or until browned and crisp on one side, then toss and repeat until the potatoes are crisp all over and can be speared easily with a fork.
Spread the mixture in the pan into an even layer and then crack the eggs on top, leaving a few inches in between them. Dot the bocconcini around the pan and then cover with a lid. Allow the eggs to steam until the whites are cooked and the cheese is melted, about 5 minutes.
Remove the lid, top the eggs with a sprinkle of black pepper, layer on the crispy prosciutto, and serve immediately with additional pesto and buttered sourdough toast if desired.