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Combine mushrooms, 3 tablespoons oil, and rosemary on a baking sheet, season with salt and freshly ground black pepper and toss to coat. Place in oven and roast until mushrooms are shrunken, and browned about, 20 minutes. Drizzle with 1 tablespoon of the balsamic and toss to coat. Scatter half of the Parmesan on the pizza. Discard rosemary and arrange mushrooms on top. Distribute ricotta across top and drizzle with remaining Parmesan and a little extra olive oil.
Bake until crust is browned, cheese is melted, about 20 minutes. Meanwhile, heat a small frying pan medium heat, and add remaining 1 tablespoon olive oil. When oil shimmers, add eggs and fry until set, bubbly and brown on the edges, and the pan is almost dry. Carefully add remaining 3 tablespoons balsamic vinegar and cook until vinegar is reduced and syrupy. Place egg on top of pizza, drizzle balsamic reduction over pizza, and serve.