Start by crisping up the prosciutto. You can skip this step and use raw prosciutto but it is amazing served crispy. Find a small oven safe bowl - or a muffin tin, and form the prosciutto into a small cup that will later hold the egg. Bake this at 350 degrees for 10 minutes, or until crisp.
While the prosciutto is baking, start work on the hash browns. Place the pesto and hash browns in a small skillet, season with salt and pepper, and stir until the pesto is evenly coating the potatoes. Cook over medium heat and flatten out the hash browns cooking one side of them. Once one side has browned, flip the hash browns and form a nest shape (a circle with a hole in the middle).
Place the crispy prosciutto into the hole in the center of the hash browns. Add the goat cheese on top of the prosciutto, and then crack the egg over the goat cheese and into the prosciutto cup. Cover the skillet and reduce the heat to medium low (use a plate if the skillet doesn't have a lid). Cook until the egg white has set, this will take about 4 minutes.
Serve immediately - tomatoes are great on the side.