In a medium-sized saucepan, heat olive oil over medium-high heat. Add red onion and cook until slightly translucent. Stir in mushrooms and cook until they release their juices, about 5 minutes. Stir in balsamic vinegar and cook for additional 2-3 minutes. Remove from heat.
Stretch out pizza dough so it's a ¼ inch thick oval shape. Transfer dough to a parchment paper lined baking sheet.
Sprinkle dough with asiago cheese, then top with prosciutto slices. Top with onion and mushroom mixture, then add gorgonzola cheese.
Pull out top oven rack and place baking sheet on top. Crack eggs into center of pizza before pushing the rack in.
Bake for 10-12 minutes, until the whites have set and cheese has melted.